GenericDecompositionOperations<T>.GeneralizedEigenvalueDecompose Method

Definition

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

Overload List

GeneralizedEigenvalueDecompose(Char, Char, Int32, Array2D<TComplex>, Array2D<TComplex>, Array1D<TComplex>, Array1D<TComplex>, Array2D<TComplex>, Array2D<TComplex>, Int32)	Computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
GeneralizedEigenvalueDecompose(Char, Char, Int32, Span2D<TComplex>, Span2D<TComplex>, Span<TComplex>, Span<TComplex>, Span2D<TComplex>, Span2D<TComplex>, Int32)	Computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
GeneralizedEigenvalueDecompose(Char, Char, Int32, Array2D<TReal>, Array2D<TReal>, Array1D<TReal>, Array1D<TReal>, Array1D<TReal>, Array2D<TReal>, Array2D<TReal>, Int32)	Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
GeneralizedEigenvalueDecompose(Char, Char, Int32, Span2D<TReal>, Span2D<TReal>, Span<TReal>, Span<TReal>, Span<TReal>, Span2D<TReal>, Span2D<TReal>, Int32)	Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Span<Complex<T>>, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32)	Computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<T>, Int32, Span<T>, Int32, Span<T>, Span<T>, Span<T>, Span<T>, Int32, Span<T>, Int32, Int32)	Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Span<Complex<T>>, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32)

Computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

public override void GeneralizedEigenvalueDecompose(
	char jobvl,
	char jobvr,
	int n,
	Span<Complex<T>> a,
	int lda,
	Span<Complex<T>> b,
	int ldb,
	Span<Complex<T>> alpha,
	Span<Complex<T>> beta,
	Span<Complex<T>> vl,
	int ldvl,
	Span<Complex<T>> vr,
	int ldvr,
	out int info
)

Visual Basic

Public Overrides Sub GeneralizedEigenvalueDecompose ( 
	jobvl As Char,
	jobvr As Char,
	n As Integer,
	a As Span(Of Complex(Of T)),
	lda As Integer,
	b As Span(Of Complex(Of T)),
	ldb As Integer,
	alpha As Span(Of Complex(Of T)),
	beta As Span(Of Complex(Of T)),
	vl As Span(Of Complex(Of T)),
	ldvl As Integer,
	vr As Span(Of Complex(Of T)),
	ldvr As Integer,
	<OutAttribute> ByRef info As Integer
)

C++/CLI

public:
virtual void GeneralizedEigenvalueDecompose(
	wchar_t jobvl, 
	wchar_t jobvr, 
	int n, 
	Span<Complex<T>> a, 
	int lda, 
	Span<Complex<T>> b, 
	int ldb, 
	Span<Complex<T>> alpha, 
	Span<Complex<T>> beta, 
	Span<Complex<T>> vl, 
	int ldvl, 
	Span<Complex<T>> vr, 
	int ldvr, 
	[OutAttribute] int% info
) override

abstract GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Span<Complex<'T>> * 
        lda : int * 
        b : Span<Complex<'T>> * 
        ldb : int * 
        alpha : Span<Complex<'T>> * 
        beta : Span<Complex<'T>> * 
        vl : Span<Complex<'T>> * 
        ldvl : int * 
        vr : Span<Complex<'T>> * 
        ldvr : int * 
        info : int byref -> unit 
override GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Span<Complex<'T>> * 
        lda : int * 
        b : Span<Complex<'T>> * 
        ldb : int * 
        alpha : Span<Complex<'T>> * 
        beta : Span<Complex<'T>> * 
        vl : Span<Complex<'T>> * 
        ldvl : int * 
        vr : Span<Complex<'T>> * 
        ldvr : int * 
        info : int byref -> unit

Parameters

jobvl Char

JOBVL is CHARACTER*1
= 'N':  do not compute the left generalized eigenvectors;
= 'V':  compute the left generalized eigenvectors.

jobvr Char

JOBVR is CHARACTER*1
= 'N':  do not compute the right generalized eigenvectors;
= 'V':  compute the right generalized eigenvectors.

n Int32

N is INTEGER
The order of the matrices A, B, VL, and VR.  N >= 0.

a Span<Complex<T>>

A is COMPLEX*16 array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.

lda Int32

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

b Span<Complex<T>>

B is COMPLEX*16 array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.

ldb Int32

LDB is INTEGER
The leading dimension of B.  LDB >= max(1,N).

alpha Span<Complex<T>>

ALPHA is COMPLEX*16 array, dimension (N)

beta Span<Complex<T>>

BETA is COMPLEX*16 array, dimension (N)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues.
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).

vl Span<Complex<T>>

VL is COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors u(j) are
stored one after another in the columns of VL, in the same
order as their eigenvalues.
Each eigenvector is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVL = 'N'.

ldvl Int32

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.

vr Span<Complex<T>>

VR is COMPLEX*16 array, dimension (LDVR,N)
If JOBVR = 'V', the right generalized eigenvectors v(j) are
stored one after another in the columns of VR, in the same
order as their eigenvalues.
Each eigenvector is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVR = 'N'.

ldvr Int32

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = 'V', LDVR >= N.

info Int32

INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
=1,...,N:
      The QZ iteration failed.  No eigenvectors have been
      calculated, but ALPHA(j) and BETA(j) should be
      correct for j=INFO+1,...,N.
> N:  =N+1: other then QZ iteration failed in DHGEQZ,
      =N+2: error return from DTGEVC.

Remarks

A generalized eigenvalue for a pair of matrices (A,B) is a scalar
lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
The right generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies
             A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues lambda(j) of (A,B) satisfies
             u(j)**H * A = lambda(j) * u(j)**H * B
where u(j)**H is the conjugate-transpose of u(j).

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

Date: April 2012

GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<T>, Int32, Span<T>, Int32, Span<T>, Span<T>, Span<T>, Span<T>, Int32, Span<T>, Int32, Int32)

Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

public override void GeneralizedEigenvalueDecompose(
	char jobvl,
	char jobvr,
	int n,
	Span<T> a,
	int lda,
	Span<T> b,
	int ldb,
	Span<T> alphar,
	Span<T> alphai,
	Span<T> beta,
	Span<T> vl,
	int ldvl,
	Span<T> vr,
	int ldvr,
	out int info
)

Visual Basic

Public Overrides Sub GeneralizedEigenvalueDecompose ( 
	jobvl As Char,
	jobvr As Char,
	n As Integer,
	a As Span(Of T),
	lda As Integer,
	b As Span(Of T),
	ldb As Integer,
	alphar As Span(Of T),
	alphai As Span(Of T),
	beta As Span(Of T),
	vl As Span(Of T),
	ldvl As Integer,
	vr As Span(Of T),
	ldvr As Integer,
	<OutAttribute> ByRef info As Integer
)

C++/CLI

public:
virtual void GeneralizedEigenvalueDecompose(
	wchar_t jobvl, 
	wchar_t jobvr, 
	int n, 
	Span<T> a, 
	int lda, 
	Span<T> b, 
	int ldb, 
	Span<T> alphar, 
	Span<T> alphai, 
	Span<T> beta, 
	Span<T> vl, 
	int ldvl, 
	Span<T> vr, 
	int ldvr, 
	[OutAttribute] int% info
) override

abstract GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Span<'T> * 
        lda : int * 
        b : Span<'T> * 
        ldb : int * 
        alphar : Span<'T> * 
        alphai : Span<'T> * 
        beta : Span<'T> * 
        vl : Span<'T> * 
        ldvl : int * 
        vr : Span<'T> * 
        ldvr : int * 
        info : int byref -> unit 
override GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Span<'T> * 
        lda : int * 
        b : Span<'T> * 
        ldb : int * 
        alphar : Span<'T> * 
        alphai : Span<'T> * 
        beta : Span<'T> * 
        vl : Span<'T> * 
        ldvl : int * 
        vr : Span<'T> * 
        ldvr : int * 
        info : int byref -> unit

Parameters

jobvl Char

JOBVL is CHARACTER*1
= 'N':  do not compute the left generalized eigenvectors;
= 'V':  compute the left generalized eigenvectors.

jobvr Char

JOBVR is CHARACTER*1
= 'N':  do not compute the right generalized eigenvectors;
= 'V':  compute the right generalized eigenvectors.

n Int32

N is INTEGER
The order of the matrices A, B, VL, and VR.  N >= 0.

a Span<T>

A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.

lda Int32

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

b Span<T>

B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.

ldb Int32

LDB is INTEGER
The leading dimension of B.  LDB >= max(1,N).

alphar Span<T>

ALPHAR is DOUBLE PRECISION array, dimension (N)

alphai Span<T>

ALPHAI is DOUBLE PRECISION array, dimension (N)

beta Span<T>

BETA is DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues.  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
alpha/beta.  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).

vl Span<T>

VL is DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVL = 'N'.

ldvl Int32

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.

vr Span<T>

VR is DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVR = 'N'.

ldvr Int32

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = 'V', LDVR >= N.

info Int32

INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
= 1,...,N:
      The QZ iteration failed.  No eigenvectors have been
      calculated, 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: error return from DTGEVC.

Remarks

A generalized eigenvalue for a pair of matrices (A,B) is a scalar
lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
of (A,B) satisfies
                 A * v(j) = lambda(j) * B * v(j).
The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
of (A,B) satisfies
                 u(j)**H * A  = lambda(j) * u(j)**H * B .
where u(j)**H is the conjugate-transpose of u(j).

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

Date: April 2012

GenericDecompositionOperations<T>.GeneralizedEigenvalueDecompose Method

Definition

Overload List

GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Span<Complex<T>>, Span<Complex<T>>, Span<Complex<T>>, Int32, Span<Complex<T>>, Int32, Int32)

Parameters

Remarks

GeneralizedEigenvalueDecompose(Char, Char, Int32, Span<T>, Int32, Span<T>, Int32, Span<T>, Span<T>, Span<T>, Span<T>, Int32, Span<T>, Int32, Int32)

Parameters

Remarks

See Also

Reference