geexp

  Signature: ([io,phys]A(n,n);int deg();scale();[io]trace();int [o]ns();int [o]info(); int [t]ipiv(n); [t]wsp(wspn))

Computes exp(t*A), the matrix exponential of a general matrix, using the irreducible rational Pade approximation to the exponential function exp(x) = r(x) = (+/-)( I + 2*(q(x)/p(x)) ), combined with scaling-and-squaring and optionally normalization of the trace. The algorithm is described in Roger B. Sidje (rbs.uq.edu.au) "EXPOKIT: Software Package for Computing Matrix Exponentials". ACM - Transactions On Mathematical Software, 24(1):130-156, 1998

     A:         On input argument matrix. On output exp(t*A).
                Use Fortran storage type.
     deg:       the degre of the diagonal Pade to be used. 
                a value of 6 is generally satisfactory. 
     scale:     time-scale (can be < 0). 
     trace:     on input, boolean value indicating whether or not perform
                a trace normalization. On output value used.
     ns:        on output number of scaling-squaring used. 
     info:      exit flag.
                      0 - no problem 
                     > 0 - Singularity in LU factorization when solving 
                     Pade approximation

  = random(5,5);
  = pdl(1);
 ->t->geexp(6,1,, ( = null), ( = null));

geexp does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

cgeexp

  Signature: (complex [io,phys]A(n,n);int deg();scale();int trace();int [o]ns();int [o]info(); int [t] ipiv(n))

Complex version of geexp. The value used for trace normalization is not returned. The algorithm is described in Roger B. Sidje (rbs@maths.uq.edu.au) "EXPOKIT: Software Package for Computing Matrix Exponentials". ACM - Transactions On Mathematical Software, 24(1):130-156, 1998

ctrsqrt

  Signature: (complex [io,phys]A(n,n);int uplo();complex [phys,o] B(n,n);int [o]info())

Root square of complex triangular matrix. Uses a recurrence of Björck and Hammarling. (See Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999. It's available at http://www.ma.man.ac.uk/~higham/pap-mf.html) If uplo is true, A is lower triangular.

ctrfun

  Signature: (complex [io]A(n,n);int uplo();complex [o] B(n,n);int [o]info(); complex [t]diag(n);SV* func)

Apply an arbitrary function to a complex triangular matrix. Uses a recurrence of Parlett. If uplo is true, A is lower triangular.

mlog

Return matrix logarithm of a square matrix.

 PDL = mlog(PDL(A))

 my $a = random(10,10);
 my $log = mlog($a);

msqrt

Return matrix square root (principal) of a square matrix.

 PDL = msqrt(PDL(A))

 my $a = random(10,10);
 my $sqrt = msqrt($a);

mexp

Return matrix exponential of a square matrix.

 PDL = mexp(PDL(A))

 my $a = random(10,10);
 my $exp = mexp($a);

mpow

Return matrix power of a square matrix.

 PDL = mpow(PDL(A), SCALAR(exponent))

 my $a = random(10,10);
 my $powered = mpow($a,2.5);

mcos

Return matrix cosine of a square matrix.

 PDL = mcos(PDL(A))

 my $a = random(10,10);
 my $cos = mcos($a);

macos

Return matrix inverse cosine of a square matrix.

 PDL = macos(PDL(A))

 my $a = random(10,10);
 my $acos = macos($a);

msin

Return matrix sine of a square matrix.

 PDL = msin(PDL(A))

 my $a = random(10,10);
 my $sin = msin($a);

masin

Return matrix inverse sine of a square matrix.

 PDL = masin(PDL(A))

 my $a = random(10,10);
 my $asin = masin($a);

mtan

Return matrix tangent of a square matrix.

 PDL = mtan(PDL(A))

 my $a = random(10,10);
 my $tan = mtan($a);

matan

Return matrix inverse tangent of a square matrix.

 PDL = matan(PDL(A))

 my $a = random(10,10);
 my $atan = matan($a);

mcot

Return matrix cotangent of a square matrix.

 PDL = mcot(PDL(A))

 my $a = random(10,10);
 my $cot = mcot($a);

macot

Return matrix inverse cotangent of a square matrix.

 PDL = macot(PDL(A))

 my $a = random(10,10);
 my $acot = macot($a);

msec

Return matrix secant of a square matrix.

 PDL = msec(PDL(A))

 my $a = random(10,10);
 my $sec = msec($a);

masec

Return matrix inverse secant of a square matrix.

 PDL = masec(PDL(A))

 my $a = random(10,10);
 my $asec = masec($a);

mcsc

Return matrix cosecant of a square matrix.

 PDL = mcsc(PDL(A))

 my $a = random(10,10);
 my $csc = mcsc($a);

macsc

Return matrix inverse cosecant of a square matrix.

 PDL = macsc(PDL(A))

 my $a = random(10,10);
 my $acsc = macsc($a);

mcosh

Return matrix hyperbolic cosine of a square matrix.

 PDL = mcosh(PDL(A))

 my $a = random(10,10);
 my $cos = mcosh($a);

macosh

Return matrix hyperbolic inverse cosine of a square matrix.

 PDL = macosh(PDL(A))

 my $a = random(10,10);
 my $acos = macosh($a);

msinh

Return matrix hyperbolic sine of a square matrix.

 PDL = msinh(PDL(A))

 my $a = random(10,10);
 my $sinh = msinh($a);

masinh

Return matrix hyperbolic inverse sine of a square matrix.

 PDL = masinh(PDL(A))

 my $a = random(10,10);
 my $asinh = masinh($a);

mtanh

Return matrix hyperbolic tangent of a square matrix.

 PDL = mtanh(PDL(A))

 my $a = random(10,10);
 my $tanh = mtanh($a);

matanh

Return matrix hyperbolic inverse tangent of a square matrix.

 PDL = matanh(PDL(A))

 my $a = random(10,10);
 my $atanh = matanh($a);

mcoth

Return matrix hyperbolic cotangent of a square matrix.

 PDL = mcoth(PDL(A))

 my $a = random(10,10);
 my $coth = mcoth($a);

macoth

Return matrix hyperbolic inverse cotangent of a square matrix.

 PDL = macoth(PDL(A))

 my $a = random(10,10);
 my $acoth = macoth($a);

msech

Return matrix hyperbolic secant of a square matrix.

 PDL = msech(PDL(A))

 my $a = random(10,10);
 my $sech = msech($a);

masech

Return matrix hyperbolic inverse secant of a square matrix.

 PDL = masech(PDL(A))

 my $a = random(10,10);
 my $asech = masech($a);

mcsch

Return matrix hyperbolic cosecant of a square matrix.

 PDL = mcsch(PDL(A))

 my $a = random(10,10);
 my $csch = mcsch($a);

macsch

Return matrix hyperbolic inverse cosecant of a square matrix.

 PDL = macsch(PDL(A))

 my $a = random(10,10);
 my $acsch = macsch($a);

mfun

Return matrix function of second argument of a square matrix. Function will be applied on a complex ndarray.

 PDL = mfun(PDL(A),'cos')

 my $a = random(10,10);
 my $fun = mfun($a,'cos');
 sub sinbycos2{
        $_[0]->set_inplace(0);
        $_[0] .= $_[0]->Csin/$_[0]->Ccos**2;
 }
 # Try diagonalization
 $fun = mfun($a, \&sinbycos2,1);
 # Now try Schur/Parlett
 $fun = mfun($a, \&sinbycos2);
 # Now with function.
 scalar msolve($a->mcos->mpow(2), $a->msin);