ZGESC2 solves a system of linear equations


           A * X = scale* RHS


 with a general N-by-N matrix A using the LU factorization with


 complete pivoting computed by ZGETC2.

N

          N is INTEGER


          The number of columns of the matrix A.

A

          A is COMPLEX*16 array, dimension (LDA, N)


          On entry, the  LU part of the factorization of the n-by-n


          matrix A computed by ZGETC2:  A = P * L * U * Q

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1, N).

RHS

          RHS is COMPLEX*16 array, dimension N.


          On entry, the right hand side vector b.


          On exit, the solution vector X.

IPIV

          IPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= i <= N, row i of the


          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= j <= N, column j of the


          matrix has been interchanged with column JPIV(j).

SCALE

          SCALE is DOUBLE PRECISION


           On exit, SCALE contains the scale factor. SCALE is chosen


           0 <= SCALE <= 1 to prevent overflow in the solution.

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Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

 ZGETC2 computes an LU factorization, using complete pivoting, of the


 n-by-n matrix A. The factorization has the form A = P * L * U * Q,


 where P and Q are permutation matrices, L is lower triangular with


 unit diagonal elements and U is upper triangular.


 This is a level 1 BLAS version of the algorithm.

N

          N is INTEGER


          The order of the matrix A. N >= 0.

A

          A is COMPLEX*16 array, dimension (LDA, N)


          On entry, the n-by-n matrix to be factored.


          On exit, the factors L and U from the factorization


          A = P*L*U*Q; the unit diagonal elements of L are not stored.


          If U(k, k) appears to be less than SMIN, U(k, k) is given the


          value of SMIN, giving a nonsingular perturbed system.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1, N).

IPIV

          IPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= i <= N, row i of the


          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= j <= N, column j of the


          matrix has been interchanged with column JPIV(j).

INFO

          INFO is INTEGER


           = 0: successful exit


           > 0: if INFO = k, U(k, k) is likely to produce overflow if


                one tries to solve for x in Ax = b. So U is perturbed


                to avoid the overflow.

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Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

 ZLANGE  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 complex matrix A.

ZLANGE

    ZLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in ZLANGE as described


          above.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.  When M = 0,


          ZLANGE is set to zero.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0.  When N = 0,


          ZLANGE is set to zero.

A

          A is COMPLEX*16 array, dimension (LDA,N)


          The m by n matrix A.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(M,1).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),


          where LWORK >= M when NORM = 'I'; otherwise, WORK is not


          referenced.

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 ZLAQGE equilibrates a general M by N matrix A using the row and


 column scaling factors in the vectors R and C.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0.

A

          A is COMPLEX*16 array, dimension (LDA,N)


          On entry, the M by N matrix A.


          On exit, the equilibrated matrix.  See EQUED for the form of


          the equilibrated matrix.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(M,1).

R

          R is DOUBLE PRECISION array, dimension (M)


          The row scale factors for A.

C

          C is DOUBLE PRECISION array, dimension (N)


          The column scale factors for A.

ROWCND

          ROWCND is DOUBLE PRECISION


          Ratio of the smallest R(i) to the largest R(i).

COLCND

          COLCND is DOUBLE PRECISION


          Ratio of the smallest C(i) to the largest C(i).

AMAX

          AMAX is DOUBLE PRECISION


          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1


          Specifies the form of equilibration that was done.


          = 'N':  No equilibration


          = 'R':  Row equilibration, i.e., A has been premultiplied by


                  diag(R).


          = 'C':  Column equilibration, i.e., A has been postmultiplied


                  by diag(C).


          = 'B':  Both row and column equilibration, i.e., A has been


                  replaced by diag(R) * A * diag(C).

  THRESH is a threshold value used to decide if row or column scaling


  should be done based on the ratio of the row or column scaling


  factors.  If ROWCND < THRESH, row scaling is done, and if


  COLCND < THRESH, column scaling is done.


  LARGE and SMALL are threshold values used to decide if row scaling


  should be done based on the absolute size of the largest matrix


  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

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 ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)


 in an upper triangular matrix pair (A, B) by an unitary equivalence


 transformation.


 (A, B) must be in generalized Schur canonical form, that is, A and


 B are both upper triangular.


 Optionally, the matrices Q and Z of generalized Schur vectors are


 updated.


        Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H


        Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

WANTQ

          WANTQ is LOGICAL


          .TRUE. : update the left transformation matrix Q;


          .FALSE.: do not update Q.

WANTZ

          WANTZ is LOGICAL


          .TRUE. : update the right transformation matrix Z;


          .FALSE.: do not update Z.

N

          N is INTEGER


          The order of the matrices A and B. N >= 0.

A

          A is COMPLEX*16 array, dimensions (LDA,N)


          On entry, the matrix A in the pair (A, B).


          On exit, the updated matrix A.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,N).

B

          B is COMPLEX*16 array, dimensions (LDB,N)


          On entry, the matrix B in the pair (A, B).


          On exit, the updated matrix B.

LDB

          LDB is INTEGER


          The leading dimension of the array B. LDB >= max(1,N).

Q

          Q is COMPLEX*16 array, dimension (LDQ,N)


          If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,


          the updated matrix Q.


          Not referenced if WANTQ = .FALSE..

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q. LDQ >= 1;


          If WANTQ = .TRUE., LDQ >= N.

Z

          Z is COMPLEX*16 array, dimension (LDZ,N)


          If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,


          the updated matrix Z.


          Not referenced if WANTZ = .FALSE..

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1;


          If WANTZ = .TRUE., LDZ >= N.

J1

          J1 is INTEGER


          The index to the first block (A11, B11).

INFO

          INFO is INTEGER


           =0:  Successful exit.


           =1:  The transformed matrix pair (A, B) would be too far


                from generalized Schur form; the problem is ill-


                conditioned.

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In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.