ManagedLapack.GeneralizedEigenvalueDecompose Method

public override void GeneralizedEigenvalueDecompose(
	char jobvl,
	char jobvr,
	int n,
	Array2D<Complex<double>> a,
	Array2D<Complex<double>> b,
	Array1D<Complex<double>> alpha,
	Array1D<Complex<double>> beta,
	Array2D<Complex<double>> vl,
	Array2D<Complex<double>> vr,
	out int info
)

Public Overrides Sub GeneralizedEigenvalueDecompose ( 
	jobvl As Char,
	jobvr As Char,
	n As Integer,
	a As Array2D(Of Complex(Of Double)),
	b As Array2D(Of Complex(Of Double)),
	alpha As Array1D(Of Complex(Of Double)),
	beta As Array1D(Of Complex(Of Double)),
	vl As Array2D(Of Complex(Of Double)),
	vr As Array2D(Of Complex(Of Double)),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void GeneralizedEigenvalueDecompose(
	wchar_t jobvl, 
	wchar_t jobvr, 
	int n, 
	Array2D<Complex<double>> a, 
	Array2D<Complex<double>> b, 
	Array1D<Complex<double>> alpha, 
	Array1D<Complex<double>> beta, 
	Array2D<Complex<double>> vl, 
	Array2D<Complex<double>> vr, 
	[OutAttribute] int% info
) override

abstract GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Array2D<Complex<float>> * 
        b : Array2D<Complex<float>> * 
        alpha : Array1D<Complex<float>> * 
        beta : Array1D<Complex<float>> * 
        vl : Array2D<Complex<float>> * 
        vr : Array2D<Complex<float>> * 
        info : int byref -> unit 
override GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Array2D<Complex<float>> * 
        b : Array2D<Complex<float>> * 
        alpha : Array1D<Complex<float>> * 
        beta : Array1D<Complex<float>> * 
        vl : Array2D<Complex<float>> * 
        vr : Array2D<Complex<float>> * 
        info : int byref -> unit 

Parameters

jobvl  Char

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

jobvr  Char

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

n  Int32

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

a  Array2D<Complex<Double>>

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

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

b  Array2D<Complex<Double>>

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

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

alpha  Array1D<Complex<Double>>

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ALPHA is COMPLEX*16 array, dimension (N)

beta  Array1D<Complex<Double>>

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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  Array2D<Complex<Double>>

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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'.

C# Copy

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

vr  Array2D<Complex<Double>>

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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'.

C# Copy

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

info  Int32

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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.

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

public override void GeneralizedEigenvalueDecompose(
	char jobvl,
	char jobvr,
	int n,
	Array2D<double> a,
	Array2D<double> b,
	Array1D<double> alphar,
	Array1D<double> alphai,
	Array1D<double> beta,
	Array2D<double> vl,
	Array2D<double> vr,
	out int info
)

Public Overrides Sub GeneralizedEigenvalueDecompose ( 
	jobvl As Char,
	jobvr As Char,
	n As Integer,
	a As Array2D(Of Double),
	b As Array2D(Of Double),
	alphar As Array1D(Of Double),
	alphai As Array1D(Of Double),
	beta As Array1D(Of Double),
	vl As Array2D(Of Double),
	vr As Array2D(Of Double),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void GeneralizedEigenvalueDecompose(
	wchar_t jobvl, 
	wchar_t jobvr, 
	int n, 
	Array2D<double> a, 
	Array2D<double> b, 
	Array1D<double> alphar, 
	Array1D<double> alphai, 
	Array1D<double> beta, 
	Array2D<double> vl, 
	Array2D<double> vr, 
	[OutAttribute] int% info
) override

abstract GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Array2D<float> * 
        b : Array2D<float> * 
        alphar : Array1D<float> * 
        alphai : Array1D<float> * 
        beta : Array1D<float> * 
        vl : Array2D<float> * 
        vr : Array2D<float> * 
        info : int byref -> unit 
override GeneralizedEigenvalueDecompose : 
        jobvl : char * 
        jobvr : char * 
        n : int * 
        a : Array2D<float> * 
        b : Array2D<float> * 
        alphar : Array1D<float> * 
        alphai : Array1D<float> * 
        beta : Array1D<float> * 
        vl : Array2D<float> * 
        vr : Array2D<float> * 
        info : int byref -> unit 

Parameters

jobvl  Char

C# Copy

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

jobvr  Char

C# Copy

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

n  Int32

C# Copy

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

a  Array2D<Double>

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

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

b  Array2D<Double>

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

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

alphar  Array1D<Double>

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ALPHAR is DOUBLE PRECISION array, dimension (N)

alphai  Array1D<Double>

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ALPHAI is DOUBLE PRECISION array, dimension (N)

beta  Array1D<Double>

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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  Array2D<Double>

C# Copy

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'.

C# Copy

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

vr  Array2D<Double>

C# Copy

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'.

C# Copy

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

info  Int32

C# Copy

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.

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

Reference