GenericDecompositionOperations<T>.SchurDecompose Method

Definition

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

Overload List

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

Computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z.

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

Computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of
Schur vectors Z.

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

Computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z.

public override void SchurDecompose(
	char jobvs,
	char sort,
	Func<Complex<T>, bool> select,
	int n,
	Span<Complex<T>> a,
	int lda,
	out int sdim,
	Span<Complex<T>> w,
	Span<Complex<T>> vs,
	int ldvs,
	out int info
)

Visual Basic

Public Overrides Sub SchurDecompose ( 
	jobvs As Char,
	sort As Char,
	select As Func(Of Complex(Of T), Boolean),
	n As Integer,
	a As Span(Of Complex(Of T)),
	lda As Integer,
	<OutAttribute> ByRef sdim As Integer,
	w As Span(Of Complex(Of T)),
	vs As Span(Of Complex(Of T)),
	ldvs As Integer,
	<OutAttribute> ByRef info As Integer
)

C++/CLI

public:
virtual void SchurDecompose(
	wchar_t jobvs, 
	wchar_t sort, 
	Func<Complex<T>, bool>^ select, 
	int n, 
	Span<Complex<T>> a, 
	int lda, 
	[OutAttribute] int% sdim, 
	Span<Complex<T>> w, 
	Span<Complex<T>> vs, 
	int ldvs, 
	[OutAttribute] int% info
) override

abstract SchurDecompose : 
        jobvs : char * 
        sort : char * 
        select : Func<Complex<'T>, bool> * 
        n : int * 
        a : Span<Complex<'T>> * 
        lda : int * 
        sdim : int byref * 
        w : Span<Complex<'T>> * 
        vs : Span<Complex<'T>> * 
        ldvs : int * 
        info : int byref -> unit 
override SchurDecompose : 
        jobvs : char * 
        sort : char * 
        select : Func<Complex<'T>, bool> * 
        n : int * 
        a : Span<Complex<'T>> * 
        lda : int * 
        sdim : int byref * 
        w : Span<Complex<'T>> * 
        vs : Span<Complex<'T>> * 
        ldvs : int * 
        info : int byref -> unit

Parameters

jobvs Char

JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

sort Char

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered:
= 'S': Eigenvalues are ordered (see SELECT).

select Func<Complex<T>, Boolean>

SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to order
to the top left of the Schur form.
IF SORT = 'N', SELECT is not referenced.
The eigenvalue W(j) is selected if SELECT(W(j)) is true.

n Int32

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

a Span<Complex<T>>

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its Schur form T.

lda Int32

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

sdim Int32

SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues for which
               SELECT is true.

w Span<Complex<T>>

W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same order that
they appear on the diagonal of the output Schur form T.

vs Span<Complex<T>>

VS is COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.

ldvs Int32

LDVS is INTEGER
The leading dimension of the array VS.  LDVS >= 1; if
JOBVS = 'V', LDVS >= N.

info Int32

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
     <= N:  the QR algorithm failed to compute all the
            eigenvalues; elements 1:ILO-1 and i+1:N of W
            contain those eigenvalues which have converged;
            if JOBVS = 'V', VS contains the matrix which
            reduces A to its partially converged Schur form.
     = N+1: the eigenvalues could not be reordered because
            some eigenvalues were too close to separate (the
            problem is very ill-conditioned);
     = N+2: after reordering, roundoff changed values of
            some complex eigenvalues so that leading
            eigenvalues in the Schur form no longer satisfy
            SELECT = .TRUE..  This could also be caused by
            underflow due to scaling.

Remarks

This gives the Schur factorization A = Z*T*(Z**H).
Optionally, it also orders the eigenvalues on the diagonal of the
Schur form so that selected eigenvalues are at the top left.
The leading columns of Z then form an orthonormal basis for the
invariant subspace corresponding to the selected eigenvalues.
A complex matrix is in Schur form if it is upper triangular.

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

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

Computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of
Schur vectors Z.

public override void SchurDecompose(
	char jobvs,
	char sort,
	Func<T, T, bool> select,
	int n,
	Span<T> a,
	int lda,
	out int sdim,
	Span<T> wr,
	Span<T> wi,
	Span<T> vs,
	int ldvs,
	out int info
)

Visual Basic

Public Overrides Sub SchurDecompose ( 
	jobvs As Char,
	sort As Char,
	select As Func(Of T, T, Boolean),
	n As Integer,
	a As Span(Of T),
	lda As Integer,
	<OutAttribute> ByRef sdim As Integer,
	wr As Span(Of T),
	wi As Span(Of T),
	vs As Span(Of T),
	ldvs As Integer,
	<OutAttribute> ByRef info As Integer
)

C++/CLI

public:
virtual void SchurDecompose(
	wchar_t jobvs, 
	wchar_t sort, 
	Func<T, T, bool>^ select, 
	int n, 
	Span<T> a, 
	int lda, 
	[OutAttribute] int% sdim, 
	Span<T> wr, 
	Span<T> wi, 
	Span<T> vs, 
	int ldvs, 
	[OutAttribute] int% info
) override

abstract SchurDecompose : 
        jobvs : char * 
        sort : char * 
        select : Func<'T, 'T, bool> * 
        n : int * 
        a : Span<'T> * 
        lda : int * 
        sdim : int byref * 
        wr : Span<'T> * 
        wi : Span<'T> * 
        vs : Span<'T> * 
        ldvs : int * 
        info : int byref -> unit 
override SchurDecompose : 
        jobvs : char * 
        sort : char * 
        select : Func<'T, 'T, bool> * 
        n : int * 
        a : Span<'T> * 
        lda : int * 
        sdim : int byref * 
        wr : Span<'T> * 
        wi : Span<'T> * 
        vs : Span<'T> * 
        ldvs : int * 
        info : int byref -> unit

Parameters

jobvs Char

JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

sort Char

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).

select Func<T, T, Boolean>

SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to sort
to the top left of the Schur form.
If SORT = 'N', SELECT is not referenced.
An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
SELECT(WR(j),WI(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 a selected complex eigenvalue may no longer
satisfy SELECT(WR(j),WI(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

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

a Span<T>

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its real Schur form T.

lda Int32

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

sdim Int32

SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
               for which SELECT is true. (Complex conjugate
               pairs for which SELECT is true for either
               eigenvalue count as 2.)

wr Span<T>

WR is DOUBLE PRECISION array, dimension (N)

wi Span<T>

WI is DOUBLE PRECISION array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues in the same order
that they appear on the diagonal of the output Schur form T.
Complex conjugate pairs of eigenvalues will appear
consecutively with the eigenvalue having the positive
imaginary part first.

vs Span<T>

VS is DOUBLE PRECISION array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.

ldvs Int32

LDVS is INTEGER
The leading dimension of the array VS.  LDVS >= 1; if
JOBVS = 'V', LDVS >= N.

info Int32

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
   <= N: the QR algorithm failed to compute all the
         eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
         contain those eigenvalues which have converged; if
         JOBVS = 'V', VS contains the matrix which reduces A
         to its partially converged Schur form.
   = N+1: the eigenvalues could not be reordered because some
         eigenvalues were too close to separate (the problem
         is very ill-conditioned);
   = N+2: after reordering, roundoff changed values of some
         complex eigenvalues so that leading eigenvalues in
         the Schur form no longer satisfy SELECT=.TRUE.  This
         could also be caused by underflow due to scaling.

Remarks

This gives the Schur factorization A = Z*T*(Z**T).
Optionally, it also orders the eigenvalues on the diagonal of the
real Schur form so that selected eigenvalues are at the top left.
The leading columns of Z then form an orthonormal basis for the
invariant subspace corresponding to the selected eigenvalues.
A matrix is in real Schur form if it is upper quasi-triangular with
1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
form
        [  a  b  ]
        [  c  a  ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

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

GenericDecompositionOperations<T>.SchurDecompose Method

Definition

Overload List

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

Parameters

Remarks

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

Parameters

Remarks

See Also

Reference