| complexHEcomputational(3) | LAPACK | complexHEcomputational(3) |
complexHEcomputational
subroutine checon (UPLO, N, A, LDA, IPIV, ANORM, RCOND,
WORK, INFO)
CHECON subroutine checon_3 (UPLO, N, A, LDA, E, IPIV, ANORM,
RCOND, WORK, INFO)
CHECON_3 subroutine checon_rook (UPLO, N, A, LDA, IPIV, ANORM,
RCOND, WORK, INFO)
CHECON_ROOK estimates the reciprocal of the condition number fort HE
matrices using factorization obtained with one of the bounded diagonal
pivoting methods (max 2 interchanges) subroutine cheequb (UPLO,
N, A, LDA, S, SCOND, AMAX, WORK, INFO)
CHEEQUB subroutine chegs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
CHEGS2 reduces a Hermitian definite generalized eigenproblem to
standard form, using the factorization results obtained from cpotrf
(unblocked algorithm). subroutine chegst (ITYPE, UPLO, N, A, LDA, B,
LDB, INFO)
CHEGST subroutine cherfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CHERFS subroutine cherfsx (UPLO, EQUED, N, NRHS, A, LDA, AF,
LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CHERFSX subroutine chetd2 (UPLO, N, A, LDA, D, E, TAU, INFO)
CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by
an unitary similarity transformation (unblocked algorithm). subroutine
chetf2 (UPLO, N, A, LDA, IPIV, INFO)
CHETF2 computes the factorization of a complex Hermitian matrix, using
the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS).
subroutine chetf2_rk (UPLO, N, A, LDA, E, IPIV, INFO)
CHETF2_RK computes the factorization of a complex Hermitian indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine chetf2_rook (UPLO, N, A, LDA,
IPIV, INFO)
CHETF2_ROOK computes the factorization of a complex Hermitian
indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting
method (unblocked algorithm). subroutine chetrd (UPLO, N, A, LDA, D,
E, TAU, WORK, LWORK, INFO)
CHETRD subroutine chetrd_2stage (VECT, UPLO, N, A, LDA, D, E,
TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
CHETRD_2STAGE subroutine chetrd_he2hb (UPLO, N, KD, A, LDA, AB,
LDAB, TAU, WORK, LWORK, INFO)
CHETRD_HE2HB subroutine chetrf (UPLO, N, A, LDA, IPIV, WORK,
LWORK, INFO)
CHETRF subroutine chetrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK,
INFO)
CHETRF_AA subroutine chetrf_rk (UPLO, N, A, LDA, E, IPIV, WORK,
LWORK, INFO)
CHETRF_RK computes the factorization of a complex Hermitian indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS3 blocked algorithm). subroutine chetrf_rook (UPLO, N, A, LDA,
IPIV, WORK, LWORK, INFO)
CHETRF_ROOK computes the factorization of a complex Hermitian
indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting
method (blocked algorithm, calling Level 3 BLAS). subroutine chetri
(UPLO, N, A, LDA, IPIV, WORK, INFO)
CHETRI subroutine chetri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK,
INFO)
CHETRI2 subroutine chetri2x (UPLO, N, A, LDA, IPIV, WORK, NB,
INFO)
CHETRI2X subroutine chetri_3 (UPLO, N, A, LDA, E, IPIV, WORK,
LWORK, INFO)
CHETRI_3 subroutine chetri_3x (UPLO, N, A, LDA, E, IPIV, WORK,
NB, INFO)
CHETRI_3X subroutine chetri_rook (UPLO, N, A, LDA, IPIV, WORK,
INFO)
CHETRI_ROOK computes the inverse of HE matrix using the factorization
obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine chetrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CHETRS subroutine chetrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
WORK, INFO)
CHETRS2 subroutine chetrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B,
LDB, INFO)
CHETRS_3 subroutine chetrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B,
LDB, WORK, LWORK, INFO)
CHETRS_AA subroutine chetrs_rook (UPLO, N, NRHS, A, LDA, IPIV,
B, LDB, INFO)
CHETRS_ROOK computes the solution to a system of linear equations A * X
= B for HE matrices using factorization obtained with one of the bounded
diagonal pivoting methods (max 2 interchanges) subroutine cla_heamv
(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite
matrix to calculate error bounds. real function cla_hercond_c (UPLO,
N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK)
CLA_HERCOND_C computes the infinity norm condition number of
op(A)*inv(diag(c)) for Hermitian indefinite matrices. real function
cla_hercond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
CLA_HERCOND_X computes the infinity norm condition number of
op(A)*diag(x) for Hermitian indefinite matrices. subroutine
cla_herfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF,
IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM,
ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB,
IGNORE_CWISE, INFO)
CLA_HERFSX_EXTENDED improves the computed solution to a system of
linear equations for Hermitian indefinite matrices by performing
extra-precise iterative refinement and provides error bounds and backward
error estimates for the solution. real function cla_herpvgrw (UPLO,
N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
CLA_HERPVGRW subroutine clahef (UPLO, N, NB, KB, A, LDA, IPIV,
W, LDW, INFO)
CLAHEF computes a partial factorization of a complex Hermitian
indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked
algorithm, calling Level 3 BLAS). subroutine clahef_rk (UPLO, N, NB,
KB, A, LDA, E, IPIV, W, LDW, INFO)
CLAHEF_RK computes a partial factorization of a complex Hermitian
indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting
method. subroutine clahef_rook (UPLO, N, NB, KB, A, LDA, IPIV, W,
LDW, INFO)
This is the group of complex computational functions for HE matrices
CHECON
Purpose:
CHECON estimates the reciprocal of the condition number of a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
ANORM
ANORM is REAL
The 1-norm of the original matrix A.
RCOND
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK
WORK is COMPLEX array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHECON_3
Purpose:
CHECON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex Hermitian matrix A using the factorization
computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver CHETRS_3.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U': Upper triangular, form is A = P*U*D*(U**H)*(P**T);
= 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by CHETRF_RK and CHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_RK or CHETRF_BK.
ANORM
ANORM is REAL
The 1-norm of the original matrix A.
RCOND
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK
WORK is COMPLEX array, dimension (2*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
CHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges)
Purpose:
CHECON_ROOK estimates the reciprocal of the condition number of a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF_ROOK.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF_ROOK.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_ROOK.
ANORM
ANORM is REAL
The 1-norm of the original matrix A.
RCOND
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK
WORK is COMPLEX array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester
CHEEQUB
Purpose:
CHEEQUB computes row and column scalings intended to equilibrate a
Hermitian matrix A (with respect to the Euclidean norm) and reduce
its condition number. The scale factors S are computed by the BIN
algorithm (see references) so that the scaled matrix B with elements
B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
the smallest possible condition number over all possible diagonal
scalings.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The N-by-N Hermitian matrix whose scaling factors are to be
computed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
S
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND
SCOND is REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.
AMAX
AMAX is REAL
Largest absolute value of any matrix element. If AMAX is
very close to overflow or very close to underflow, the
matrix should be scaled.
WORK
WORK is COMPLEX array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
References:
CHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm).
Purpose:
CHEGS2 reduces a complex Hermitian-definite generalized
eigenproblem to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H *A*L.
B must have been previously factorized as U**H *U or L*L**H by ZPOTRF.
Parameters:
ITYPE is INTEGER
= 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
= 2 or 3: compute U*A*U**H or L**H *A*L.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored, and how B has been factorized.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is COMPLEX array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by CPOTRF.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHEGST
Purpose:
CHEGST reduces a complex Hermitian-definite generalized
eigenproblem to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L.
B must have been previously factorized as U**H*U or L*L**H by CPOTRF.
Parameters:
ITYPE is INTEGER
= 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
= 2 or 3: compute U*A*U**H or L**H*A*L.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored and B is factored as
U**H*U;
= 'L': Lower triangle of A is stored and B is factored as
L*L**H.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is COMPLEX array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by CPOTRF.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHERFS
Purpose:
CHERFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, and
provides error bounds and backward error estimates for the solution.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**H or
A = L*D*L**H as computed by CHETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
B
B is COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CHETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK
WORK is COMPLEX array, dimension (2*N)
RWORK
RWORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHERFSX
Purpose:
CHERFSX improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, and
provides error bounds and backward error estimates for the
solution. In addition to normwise error bound, the code provides
maximum componentwise error bound if possible. See comments for
ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED and S
below. In this case, the solution and error bounds returned are
for the original unequilibrated system.
Some optional parameters are bundled in the PARAMS array. These
settings determine how refinement is performed, but often the
defaults are acceptable. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine. This is needed to compute
the solution and error bounds correctly.
= 'N': No equilibration
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
The right hand side B has been changed accordingly.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular
part of the matrix A, and the strictly lower triangular
part of A is not referenced. If UPLO = 'L', the leading
N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or A =
L*D*L**T as computed by SSYTRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF.
S
S is REAL array, dimension (N)
The scale factors for A. If EQUED = 'Y', A is multiplied on
the left and right by diag(S). S is an input argument if FACT =
'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
= 'Y', each element of S must be positive. If S is output, each
element of S is a power of the radix. If S is input, each element
of S should be a power of the radix to ensure a reliable solution
and error estimates. Scaling by powers of the radix does not cause
rounding errors unless the result underflows or overflows.
Rounding errors during scaling lead to refining with a matrix that
is not equivalent to the input matrix, producing error estimates
that may not be reliable.
B
B is COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND
RCOND is REAL
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
BERR
BERR is REAL array, dimension (NRHS)
Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).
N_ERR_BNDS
N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.
ERR_BNDS_NORM
ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP
ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If .LE. 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is .LT. 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0
= 0.0 : No refinement is performed, and no error bounds are
computed.
= 1.0 : Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm. Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)
WORK
WORK is COMPLEX array, dimension (2*N)
RWORK
RWORK is REAL array, dimension (2*N)
INFO
INFO is INTEGER
= 0: Successful exit. The solution to every right-hand side is
guaranteed.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
Purpose:
CHETD2 reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q**H * A * Q = T.
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
D
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU
TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS).
Purpose:
CHETF2 computes the factorization of a complex Hermitian matrix A
using the Bunch-Kaufman diagonal pivoting method:
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**H is the conjugate transpose of U, and D is
Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.
If UPLO = 'L':
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
09-29-06 - patch from
Bobby Cheng, MathWorks
Replace l.210 and l.392
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
If UPLO = 'U', then A = U*D*U**H, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
CHETF2_RK computes the factorization of a complex Hermitian matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
Purpose:
CHETF2_ROOK computes the factorization of a complex Hermitian matrix A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**H is the conjugate transpose of U, and D is
Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
If UPLO = 'U', then A = U*D*U**H, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
Contributors:
November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
CHETRD
Purpose:
CHETRD reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q**H * A * Q = T.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
D
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU
TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
CHETRD_2STAGE
Purpose:
CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q1**H Q2**H* A * Q2 * Q1 = T.
Parameters:
VECT is CHARACTER*1
= 'N': No need for the Housholder representation,
in particular for the second stage (Band to
tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
= 'V': the Householder representation is needed to
either generate Q1 Q2 or to apply Q1 Q2,
then LHOUS2 is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the band superdiagonal
of A are overwritten by the corresponding elements of the
internal band-diagonal matrix AB, and the elements above
the KD superdiagonal, with the array TAU, represent the unitary
matrix Q1 as a product of elementary reflectors; if UPLO
= 'L', the diagonal and band subdiagonal of A are over-
written by the corresponding elements of the internal band-diagonal
matrix AB, and the elements below the KD subdiagonal, with
the array TAU, represent the unitary matrix Q1 as a product
of elementary reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
D
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E
E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T.
TAU
TAU is COMPLEX array, dimension (N-KD)
The scalar factors of the elementary reflectors of
the first stage (see Further Details).
HOUS2
HOUS2 is COMPLEX array, dimension LHOUS2, that
store the Householder representation of the stage2
band to tridiagonal.
LHOUS2
LHOUS2 is INTEGER
The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension)
If LWORK = -1, or LHOUS2=-1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS2 array, returns
this value as the first entry of the HOUS2 array, and no error
message related to LHOUS2 is issued by XERBLA.
LHOUS2 = MAX(1, dimension) where
dimension = 4*N if VECT='N'
not available now if VECT='H'
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS2=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196
CHETRD_HE2HB
Purpose:
CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
band-diagonal form AB by a unitary similarity transformation:
Q**H * A * Q = AB.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals of the reduced matrix if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
The reduced matrix is stored in the array AB.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AB
AB is COMPLEX array, dimension (LDAB,N)
On exit, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
TAU
TAU is COMPLEX array, dimension (N-KD)
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, or if LWORK=-1,
WORK(1) returns the size of LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK which should be calculated
by a workspace query. LWORK = MAX(1, LWORK_QUERY)
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
where FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice otherwise
putting LWORK=-1 will provide the size of WORK.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
A(i,i+kd+1:n), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(k), where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
A(i+kd+2:n,i), and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( ab ab/v1 v1 v1 v1 ) ( ab )
( ab ab/v2 v2 v2 ) ( ab/v1 ab )
( ab ab/v3 v3 ) ( v1 ab/v2 ab )
( ab ab/v4 ) ( v1 v2 ab/v3 ab )
( ab ) ( v1 v2 v3 ab/v4 ab )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i)..fi
CHETRF
Purpose:
CHETRF computes the factorization of a complex Hermitian matrix A
using the Bunch-Kaufman diagonal pivoting method. The form of the
factorization is
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
If UPLO = 'U', then A = U*D*U**H, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**H, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
CHETRF_AA
Purpose:
CHETRF_AA computes the factorization of a complex hermitian matrix A
using the Aasen's algorithm. The form of the factorization is
A = U*T*U**H or A = L*T*L**H
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a hermitian tridiagonal matrix.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the tridiagonal matrix is stored in the diagonals
and the subdiagonals of A just below (or above) the diagonals,
and L is stored below (or above) the subdiaonals, when UPLO
is 'L' (or 'U').
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of WORK. LWORK >= 2*N. For optimum performance
LWORK >= N*(1+NB), where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
Purpose:
CHETRF_RK computes the factorization of a complex Hermitian matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
For more information see Further Details section.
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
WORK
WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned
by ILAENV.
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued
by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
TODO: put correct description
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
CHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
Purpose:
CHETRF_ROOK computes the factorization of a comlex Hermitian matrix A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
The form of the factorization is
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK)).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
Contributors:
June 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
CHETRI
Purpose:
CHETRI computes the inverse of a complex Hermitian indefinite matrix
A using the factorization A = U*D*U**H or A = L*D*L**H computed by
CHETRF.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (Hermitian) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
WORK
WORK is COMPLEX array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRI2
Purpose:
CHETRI2 computes the inverse of a COMPLEX hermitian indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
CHETRF. CHETRI2 set the LEADING DIMENSION of the workspace
before calling CHETRI2X that actually computes the inverse.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the NB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NB structure of D
as determined by CHETRF.
WORK
WORK is COMPLEX array, dimension (N+NB+1)*(NB+3)
LWORK
LWORK is INTEGER
The dimension of the array WORK.
WORK is size >= (N+NB+1)*(NB+3)
If LWORK = -1, then a workspace query is assumed; the routine
calculates:
- the optimal size of the WORK array, returns
this value as the first entry of the WORK array,
- and no error message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRI2X
Purpose:
CHETRI2X computes the inverse of a complex Hermitian indefinite matrix
A using the factorization A = U*D*U**H or A = L*D*L**H computed by
CHETRF.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the NNB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CHETRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NNB structure of D
as determined by CHETRF.
WORK
WORK is COMPLEX array, dimension (N+NB+1,NB+3)
NB
NB is INTEGER
Block size
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRI_3
Purpose:
CHETRI_3 computes the inverse of a complex Hermitian indefinite
matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
CHETRI_3 sets the leading dimension of the workspace before calling
CHETRI_3X that actually computes the inverse. This is the blocked
version of the algorithm, calling Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by CHETRF_RK and CHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the Hermitian inverse of the original
matrix.
If UPLO = 'U': the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = 'L': the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_RK or CHETRF_BK.
WORK
WORK is COMPLEX array, dimension (N+NB+1)*(NB+3).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of WORK. LWORK >= (N+NB+1)*(NB+3).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the optimal
size of the WORK array, returns this value as the first
entry of the WORK array, and no error message related to
LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
CHETRI_3X
Purpose:
CHETRI_3X computes the inverse of a complex Hermitian indefinite
matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by CHETRF_RK and CHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the Hermitian inverse of the original
matrix.
If UPLO = 'U': the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = 'L': the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_RK or CHETRF_BK.
WORK
WORK is COMPLEX array, dimension (N+NB+1,NB+3).
NB
NB is INTEGER
Block size.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
CHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
Purpose:
CHETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix
A using the factorization A = U*D*U**H or A = L*D*L**H computed by
CHETRF_ROOK.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CHETRF_ROOK.
On exit, if INFO = 0, the (Hermitian) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_ROOK.
WORK
WORK is COMPLEX array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
CHETRS
Purpose:
CHETRS solves a system of linear equations A*X = B with a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRS2
Purpose:
CHETRS2 solves a system of linear equations A*X = B with a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF and converted by CSYCONV.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
WORK
WORK is COMPLEX array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRS_3
Purpose:
CHETRS_3 solves a system of linear equations A * X = B with a complex
Hermitian matrix A using the factorization computed
by CHETRF_RK or CHETRF_BK:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This algorithm is using Level 3 BLAS.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U': Upper triangular, form is A = P*U*D*(U**H)*(P**T);
= 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by CHETRF_RK and CHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_RK or CHETRF_BK.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
CHETRS_AA
Purpose:
CHETRS_AA solves a system of linear equations A*X = B with a complex
hermitian matrix A using the factorization A = U*T*U**H or
A = L*T*L**H computed by CHETRF_AA.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*T*U**H;
= 'L': Lower triangular, form is A = L*T*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
Details of factors computed by CHETRF_AA.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by CHETRF_AA.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
WORK
WORK is DOUBLE array, dimension (MAX(1,LWORK))
LWORK
LWORK is INTEGER, LWORK >= MAX(1,3*N-2).
aram[out]
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE matrices using factorization obtained with one of the bounded diagonal pivoting methods (max 2 interchanges)
Purpose:
CHETRS_ROOK solves a system of linear equations A*X = B with a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF_ROOK.
Parameters:
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF_ROOK.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_ROOK.
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.
Purpose:
CLA_SYAMV performs the matrix-vector operation
y := alpha*abs(A)*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an
n by n symmetric matrix.
This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
"symbolically" zero components are not perturbed. A zero
entry is considered "symbolic" if all multiplications involved
in computing that entry have at least one zero multiplicand.
Parameters:
UPLO is INTEGER
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
follows:
UPLO = BLAS_UPPER Only the upper triangular part of A
is to be referenced.
UPLO = BLAS_LOWER Only the lower triangular part of A
is to be referenced.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is REAL .
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A
A is COMPLEX array, dimension ( LDA, n ).
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.
LDA
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, n ).
Unchanged on exit.
X
X is COMPLEX array, dimension
( 1 + ( n - 1 )*abs( INCX ) )
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.
INCX
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
BETA
BETA is REAL .
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.
Y
Y is REAL array, dimension
( 1 + ( n - 1 )*abs( INCY ) )
Before entry with BETA non-zero, the incremented array Y
must contain the vector y. On exit, Y is overwritten by the
updated vector y.
INCY
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
Level 2 Blas routine.
-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
-- Modified for the absolute-value product, April 2006
Jason Riedy, UC Berkeley
CLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.
Purpose:
CLA_HERCOND_C computes the infinity norm condition number of
op(A) * inv(diag(C)) where C is a REAL vector.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
C
C is REAL array, dimension (N)
The vector C in the formula op(A) * inv(diag(C)).
CAPPLY
CAPPLY is LOGICAL
If .TRUE. then access the vector C in the formula above.
INFO
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK
WORK is COMPLEX array, dimension (2*N).
Workspace.
RWORK
RWORK is REAL array, dimension (N).
Workspace.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.
Purpose:
CLA_HERCOND_X computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX vector.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
X
X is COMPLEX array, dimension (N)
The vector X in the formula op(A) * diag(X).
INFO
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK
WORK is COMPLEX array, dimension (2*N).
Workspace.
RWORK
RWORK is REAL array, dimension (N).
Workspace.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
Purpose:
CLA_HERFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by CHERFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only resonsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters:
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and
P = 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X', 'E': Extra
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is REAL array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
B
B is COMPLEX array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is COMPLEX array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by CHETRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is REAL array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by CLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
ERR_BNDS_NORM
ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP
ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is COMPLEX array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is REAL array, dimension (N)
Workspace.
DY
DY is COMPLEX array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is COMPLEX array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
RCOND
RCOND is REAL
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For 'aggressive' set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
RTHRESH
RTHRESH is REAL
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is REAL
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we definte as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to CLA_HERFSX_EXTENDED had an illegal
value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CLA_HERPVGRW
Purpose:
CLA_HERPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable.
Parameters:
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
INFO
INFO is INTEGER
The value of INFO returned from SSYTRF, .i.e., the pivot in
column INFO is exactly 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
WORK
WORK is REAL array, dimension (2*N)
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
Purpose:
CLAHEF computes a partial factorization of a complex Hermitian
matrix A using the Bunch-Kaufman diagonal pivoting method. The
partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**H denotes the conjugate transpose of U.
CLAHEF is an auxiliary routine called by CHETRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
NB
NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.
KB
KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.
If UPLO = 'L':
Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.
W
W is COMPLEX array, dimension (LDW,NB)
LDW
LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
Purpose:
CLAHEF_RK computes a partial factorization of a complex Hermitian
matrix A using the bounded Bunch-Kaufman (rook) diagonal
pivoting method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L',
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
CLAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
NB
NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.
KB
KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,N-KB+1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,N-KB+1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,N-KB+1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,1:KB).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
W
W is COMPLEX array, dimension (LDW,NB)
LDW
LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
Purpose:
CLAHEF_ROOK computes a partial factorization of a complex Hermitian
matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting
method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**H U22**H )
A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**H denotes the conjugate transpose of U.
CLAHEF_ROOK is an auxiliary routine called by CHETRF_ROOK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
Parameters:
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
NB
NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.
KB
KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
W
W is COMPLEX array, dimension (LDW,NB)
LDW
LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
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