GenericDecompositionOperations<T>.GeneralizedSchurDecompose Method

Definition

Namespace: Numerics.NET.LinearAlgebra.Implementation
Assembly: Numerics.NET.Generic (in Numerics.NET.Generic.dll) Version: 9.0.3

Overload List

GeneralizedSchurDecompose(Char, Char, Char, Func<Complex<T>, Complex<T>, Boolean>, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32, Span<Complex<T>>, Span<Complex<T>>, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32)

C# 
Computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur
form (S, T), and optionally left and/or right Schur vectors (VSL
and VSR).

GeneralizedSchurDecompose(Char, Char, Char, Func<T, T, T, Boolean>, Int32, Span<T>, Int32, Span<T>, Int32, Int32, Span<T>, Span<T>, Span<T>, Span<T>, Int32, Span<T>, Int32, Int32)

C# 
Computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR).

GeneralizedSchurDecompose(Char, Char, Char, Func<Complex<T>, Complex<T>, Boolean>, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32, Span<Complex<T>>, Span<Complex<T>>, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32)

C# 
Computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur
form (S, T), and optionally left and/or right Schur vectors (VSL
and VSR).

C# 
public override void GeneralizedSchurDecompose(
	char jobvsl,
	char jobvsr,
	char sort,
	Func<Complex<T>, Complex<T>, bool> selctg,
	int n,
	Span<Complex<T>> a,
	int lda,
	Span<Complex<T>> b,
	int ldb,
	out int sdim,
	Span<Complex<T>> alpha,
	Span<Complex<T>> beta,
	Span<Complex<T>> vsl,
	int ldvsl,
	Span<Complex<T>> vsr,
	int ldvsr,
	out int info
)

Parameters

jobvsl  Char
C# 
JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors.
jobvsr  Char
C# 
JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors.
sort  Char
C# 
SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N':  Eigenvalues are not ordered;
= 'S':  Eigenvalues are ordered (see SELCTG).
selctg  Func<Complex<T>, Complex<T>, Boolean>
C# 
SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue ALPHA(j)/BETA(j) is selected if
SELCTG(ALPHA(j),BETA(j)) is true.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned), in this
case INFO is set to N+2 (See INFO below).
n  Int32
C# 
N is INTEGER
The order of the matrices A, B, VSL, and VSR.  N >= 0.
a  Span<Complex<T>>
C# 
A is COMPLEX*16 array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
lda  Int32
C# 
LDA is INTEGER
The leading dimension of A.  LDA >= max(1,N).
b  Span<Complex<T>>
C# 
B is COMPLEX*16 array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
ldb  Int32
C# 
LDB is INTEGER
The leading dimension of B.  LDB >= max(1,N).
sdim  Int32
C# 
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.
alpha  Span<Complex<T>>
C# 
ALPHA is COMPLEX*16 array, dimension (N)
beta  Span<Complex<T>>
C# 
BETA is COMPLEX*16 array, dimension (N)
On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),
j=1,...,N  are the diagonals of the complex Schur form (A,B)
output by ZGGES. The  BETA(j) will be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero.  Thus, the user
should avoid naively computing the ratio alpha/beta.
However, ALPHA will be always less than and usually
comparable with norm(A) in magnitude, and BETA always less
than and usually comparable with norm(B).
vsl  Span<Complex<T>>
C# 
VSL is COMPLEX*16 array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
ldvsl  Int32
C# 
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >= 1, and
if JOBVSL = 'V', LDVSL >= N.
vsr  Span<Complex<T>>
C# 
VSR is COMPLEX*16 array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
ldvsr  Int32
C# 
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
info  Int32
C# 
INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
=1,...,N:
      The QZ iteration failed.  (A,B) are not in Schur
      form, but ALPHA(j) and BETA(j) should be correct for
      j=INFO+1,...,N.
> N:  =N+1: other than QZ iteration failed in ZHGEQZ
      =N+2: after reordering, roundoff changed values of
            some complex eigenvalues so that leading
            eigenvalues in the Generalized Schur form no
            longer satisfy SELCTG=.TRUE.  This could also
            be caused due to scaling.
      =N+3: reordering failed in ZTGSEN.

Remarks

C# 
This gives the generalized Schur factorization
        (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
where (VSR)**H is the conjugate-transpose of VSR.
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
triangular matrix S and the upper triangular matrix T. The leading
columns of VSL and VSR then form an unitary basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
ZGGEV instead, which is faster.)
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
usually represented as the pair (alpha,beta), as there is a
reasonable interpretation for beta=0, and even for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if S
and T are upper triangular and, in addition, the diagonal elements
of T are non-negative real numbers.

Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.

GeneralizedSchurDecompose(Char, Char, Char, Func<T, T, T, Boolean>, Int32, Span<T>, Int32, Span<T>, Int32, Int32, Span<T>, Span<T>, Span<T>, Span<T>, Int32, Span<T>, Int32, Int32)

C# 
Computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR).

C# 
public override void GeneralizedSchurDecompose(
	char jobvsl,
	char jobvsr,
	char sort,
	Func<T, T, T, bool> selctg,
	int n,
	Span<T> a,
	int lda,
	Span<T> b,
	int ldb,
	out int sdim,
	Span<T> alphar,
	Span<T> alphai,
	Span<T> beta,
	Span<T> vsl,
	int ldvsl,
	Span<T> vsr,
	int ldvsr,
	out int info
)

Parameters

jobvsl  Char
C# 
JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors.
jobvsr  Char
C# 
JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors.
sort  Char
C# 
SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N':  Eigenvalues are not ordered;
= 'S':  Eigenvalues are ordered (see SELCTG);
selctg  Func<T, T, T, Boolean>
C# 
SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
in this case.
n  Int32
C# 
N is INTEGER
The order of the matrices A, B, VSL, and VSR.  N >= 0.
a  Span<T>
C# 
A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
lda  Int32
C# 
LDA is INTEGER
The leading dimension of A.  LDA >= max(1,N).
b  Span<T>
C# 
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
ldb  Int32
C# 
LDB is INTEGER
The leading dimension of B.  LDB >= max(1,N).
sdim  Int32
C# 
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.  (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
alphar  Span<T>
C# 
ALPHAR is DOUBLE PRECISION array, dimension (N)
alphai  Span<T>
C# 
ALPHAI is DOUBLE PRECISION array, dimension (N)
beta  Span<T>
C# 
BETA is DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
and  BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
may easily over- or underflow, and BETA(j) may even be zero.
Thus, the user should avoid naively computing the ratio.
However, ALPHAR and ALPHAI will be always less than and
usually comparable with norm(A) in magnitude, and BETA always
less than and usually comparable with norm(B).
vsl  Span<T>
C# 
VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
ldvsl  Int32
C# 
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
vsr  Span<T>
C# 
VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
ldvsr  Int32
C# 
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
info  Int32
C# 
INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
= 1,...,N:
      The QZ iteration failed.  (A,B) are not in Schur
      form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
      be correct for j=INFO+1,...,N.
> N:  =N+1: other than QZ iteration failed in DHGEQZ.
      =N+2: after reordering, roundoff changed values of
            some complex eigenvalues so that leading
            eigenvalues in the Generalized Schur form no
            longer satisfy SELCTG=.TRUE.  This could also
            be caused due to scaling.
      =N+3: reordering failed in DTGSEN.

Remarks

C# 
This gives the generalized Schur factorization
         (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T.The
leading columns of VSL and VSR then form an orthonormal basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
DGGEV instead, which is faster.)
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
usually represented as the pair (alpha,beta), as there is a
reasonable interpretation for beta=0 or both being zero.
A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
        [  a  0  ]
        [  0  b  ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.

Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.

See Also