DecompositionOperations<TReal, TComplex>.GeneralizedSingularValueDecompose Method

public void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Array2D<TReal> a,
	Array2D<TReal> b,
	Array1D<TReal> alpha,
	Array1D<TReal> beta,
	Array2D<TReal> u,
	Array2D<TReal> v,
	Array2D<TReal> q,
	Array1D<int> ipiv,
	out int info
)

Public Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Array2D(Of TReal),
	b As Array2D(Of TReal),
	alpha As Array1D(Of TReal),
	beta As Array1D(Of TReal),
	u As Array2D(Of TReal),
	v As Array2D(Of TReal),
	q As Array2D(Of TReal),
	ipiv As Array1D(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Array2D<TReal> a, 
	Array2D<TReal> b, 
	Array1D<TReal> alpha, 
	Array1D<TReal> beta, 
	Array2D<TReal> u, 
	Array2D<TReal> v, 
	Array2D<TReal> q, 
	Array1D<int> ipiv, 
	[OutAttribute] int% info
)

member GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Array2D<'TReal> * 
        b : Array2D<'TReal> * 
        alpha : Array1D<'TReal> * 
        beta : Array1D<'TReal> * 
        u : Array2D<'TReal> * 
        v : Array2D<'TReal> * 
        q : Array2D<'TReal> * 
        ipiv : Array1D<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

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JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.

jobv  Char

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JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.

jobq  Char

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JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

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M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

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N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

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P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

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K is INTEGER

l  Int32

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L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.

a  Array2D<TReal>

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A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

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

b  Array2D<TReal>

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B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0.
See Purpose for details.

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

alpha  Array1D<TReal>

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ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Array1D<TReal>

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BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Array2D<TReal>

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U is DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced.

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LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Array2D<TReal>

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V is DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced.

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LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Array2D<TReal>

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Q is DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.

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LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Array1D<Int32>

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IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(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.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine DTGSJA.

Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the orthogonal
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
also equal to the CS decomposition of A and B. Furthermore, the GSVD
can be used to derive the solution of the eigenvalue problem:
                     A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                     X = Q*( I   0    )
                           ( 0 inv(R) ).

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

DGGSVD3 replaces the deprecated subroutine DGGSVD.

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

Date: August 2015

public void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Array2D<TComplex> a,
	Array2D<TComplex> b,
	Array1D<TReal> alpha,
	Array1D<TReal> beta,
	Array2D<TComplex> u,
	Array2D<TComplex> v,
	Array2D<TComplex> q,
	Array1D<int> ipiv,
	out int info
)

Public Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Array2D(Of TComplex),
	b As Array2D(Of TComplex),
	alpha As Array1D(Of TReal),
	beta As Array1D(Of TReal),
	u As Array2D(Of TComplex),
	v As Array2D(Of TComplex),
	q As Array2D(Of TComplex),
	ipiv As Array1D(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Array2D<TComplex> a, 
	Array2D<TComplex> b, 
	Array1D<TReal> alpha, 
	Array1D<TReal> beta, 
	Array2D<TComplex> u, 
	Array2D<TComplex> v, 
	Array2D<TComplex> q, 
	Array1D<int> ipiv, 
	[OutAttribute] int% info
)

member GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Array2D<'TComplex> * 
        b : Array2D<'TComplex> * 
        alpha : Array1D<'TReal> * 
        beta : Array1D<'TReal> * 
        u : Array2D<'TComplex> * 
        v : Array2D<'TComplex> * 
        q : Array2D<'TComplex> * 
        ipiv : Array1D<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

C# Copy

JOBU is CHARACTER*1
= 'U':  Unitary matrix U is computed;
= 'N':  U is not computed.

jobv  Char

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JOBV is CHARACTER*1
= 'V':  Unitary matrix V is computed;
= 'N':  V is not computed.

jobq  Char

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JOBQ is CHARACTER*1
= 'Q':  Unitary matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

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M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

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N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

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P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

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K is INTEGER

l  Int32

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L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**H,B**H)**H.

a  Array2D<TComplex>

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A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

C# Copy

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

b  Array2D<TComplex>

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B is COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of the triangular matrix R if
M-K-L < 0.  See Purpose for details.

C# Copy

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

alpha  Array1D<TReal>

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ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Array1D<TReal>

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BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Array2D<TComplex>

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U is COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary matrix U.
If JOBU = 'N', U is not referenced.

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LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Array2D<TComplex>

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V is COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary matrix V.
If JOBV = 'N', V is not referenced.

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LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Array2D<TComplex>

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Q is COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
If JOBQ = 'N', Q is not referenced.

C# Copy

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Array1D<Int32>

C# Copy

IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(N).

info  Int32

C# Copy

INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine ZTGSJA.

Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
matrices and of the following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the unitary
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**H.
If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
equal to the CS decomposition of A and B. Furthermore, the GSVD can
be used to derive the solution of the eigenvalue problem:
                     A**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                      X = Q*(  I   0    )
                            (  0 inv(R) )

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

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

Date: August 2015

public void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Span2D<TReal> a,
	Span2D<TReal> b,
	Span<TReal> alpha,
	Span<TReal> beta,
	Span2D<TReal> u,
	Span2D<TReal> v,
	Span2D<TReal> q,
	Span<int> ipiv,
	out int info
)

Public Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Span2D(Of TReal),
	b As Span2D(Of TReal),
	alpha As Span(Of TReal),
	beta As Span(Of TReal),
	u As Span2D(Of TReal),
	v As Span2D(Of TReal),
	q As Span2D(Of TReal),
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Span2D<TReal> a, 
	Span2D<TReal> b, 
	Span<TReal> alpha, 
	Span<TReal> beta, 
	Span2D<TReal> u, 
	Span2D<TReal> v, 
	Span2D<TReal> q, 
	Span<int> ipiv, 
	[OutAttribute] int% info
)

member GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Span2D<'TReal> * 
        b : Span2D<'TReal> * 
        alpha : Span<'TReal> * 
        beta : Span<'TReal> * 
        u : Span2D<'TReal> * 
        v : Span2D<'TReal> * 
        q : Span2D<'TReal> * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

C# Copy

JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.

jobv  Char

C# Copy

JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.

jobq  Char

C# Copy

JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

C# Copy

M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

C# Copy

N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

C# Copy

P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

C# Copy

K is INTEGER

l  Int32

C# Copy

L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.

a  Span2D<TReal>

C# Copy

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

C# Copy

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

b  Span2D<TReal>

C# Copy

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0.
See Purpose for details.

C# Copy

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

alpha  Span<TReal>

C# Copy

ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Span<TReal>

C# Copy

BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Span2D<TReal>

C# Copy

U is DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced.

C# Copy

LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Span2D<TReal>

C# Copy

V is DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced.

C# Copy

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Span2D<TReal>

C# Copy

Q is DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.

C# Copy

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Span<Int32>

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IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(N).

info  Int32

C# Copy

INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine DTGSJA.

Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the orthogonal
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
also equal to the CS decomposition of A and B. Furthermore, the GSVD
can be used to derive the solution of the eigenvalue problem:
                     A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                     X = Q*( I   0    )
                           ( 0 inv(R) ).

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

DGGSVD3 replaces the deprecated subroutine DGGSVD.

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

Date: August 2015

public void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Span2D<TComplex> a,
	Span2D<TComplex> b,
	Span<TReal> alpha,
	Span<TReal> beta,
	Span2D<TComplex> u,
	Span2D<TComplex> v,
	Span2D<TComplex> q,
	Span<int> ipiv,
	out int info
)

Public Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Span2D(Of TComplex),
	b As Span2D(Of TComplex),
	alpha As Span(Of TReal),
	beta As Span(Of TReal),
	u As Span2D(Of TComplex),
	v As Span2D(Of TComplex),
	q As Span2D(Of TComplex),
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Span2D<TComplex> a, 
	Span2D<TComplex> b, 
	Span<TReal> alpha, 
	Span<TReal> beta, 
	Span2D<TComplex> u, 
	Span2D<TComplex> v, 
	Span2D<TComplex> q, 
	Span<int> ipiv, 
	[OutAttribute] int% info
)

member GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Span2D<'TComplex> * 
        b : Span2D<'TComplex> * 
        alpha : Span<'TReal> * 
        beta : Span<'TReal> * 
        u : Span2D<'TComplex> * 
        v : Span2D<'TComplex> * 
        q : Span2D<'TComplex> * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

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JOBU is CHARACTER*1
= 'U':  Unitary matrix U is computed;
= 'N':  U is not computed.

jobv  Char

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JOBV is CHARACTER*1
= 'V':  Unitary matrix V is computed;
= 'N':  V is not computed.

jobq  Char

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JOBQ is CHARACTER*1
= 'Q':  Unitary matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

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M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

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N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

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P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

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K is INTEGER

l  Int32

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L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**H,B**H)**H.

a  Span2D<TComplex>

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A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

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

b  Span2D<TComplex>

C# Copy

B is COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of the triangular matrix R if
M-K-L < 0.  See Purpose for details.

C# Copy

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

alpha  Span<TReal>

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ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Span<TReal>

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BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Span2D<TComplex>

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U is COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary matrix U.
If JOBU = 'N', U is not referenced.

C# Copy

LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Span2D<TComplex>

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V is COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary matrix V.
If JOBV = 'N', V is not referenced.

C# Copy

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Span2D<TComplex>

C# Copy

Q is COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
If JOBQ = 'N', Q is not referenced.

C# Copy

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Span<Int32>

C# Copy

IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(N).

info  Int32

C# Copy

INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine ZTGSJA.

Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
matrices and of the following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the unitary
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**H.
If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
equal to the CS decomposition of A and B. Furthermore, the GSVD can
be used to derive the solution of the eigenvalue problem:
                     A**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                      X = Q*(  I   0    )
                            (  0 inv(R) )

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

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

Date: August 2015

public abstract void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Span<TReal> a,
	int lda,
	Span<TReal> b,
	int ldb,
	Span<TReal> alpha,
	Span<TReal> beta,
	Span<TReal> u,
	int ldu,
	Span<TReal> v,
	int ldv,
	Span<TReal> q,
	int ldq,
	Span<int> ipiv,
	out int info
)

Public MustOverride Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Span(Of TReal),
	lda As Integer,
	b As Span(Of TReal),
	ldb As Integer,
	alpha As Span(Of TReal),
	beta As Span(Of TReal),
	u As Span(Of TReal),
	ldu As Integer,
	v As Span(Of TReal),
	ldv As Integer,
	q As Span(Of TReal),
	ldq As Integer,
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Span<TReal> a, 
	int lda, 
	Span<TReal> b, 
	int ldb, 
	Span<TReal> alpha, 
	Span<TReal> beta, 
	Span<TReal> u, 
	int ldu, 
	Span<TReal> v, 
	int ldv, 
	Span<TReal> q, 
	int ldq, 
	Span<int> ipiv, 
	[OutAttribute] int% info
) abstract

abstract GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Span<'TReal> * 
        lda : int * 
        b : Span<'TReal> * 
        ldb : int * 
        alpha : Span<'TReal> * 
        beta : Span<'TReal> * 
        u : Span<'TReal> * 
        ldu : int * 
        v : Span<'TReal> * 
        ldv : int * 
        q : Span<'TReal> * 
        ldq : int * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

C# Copy

JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.

jobv  Char

C# Copy

JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.

jobq  Char

C# Copy

JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

C# Copy

M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

C# Copy

N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

C# Copy

P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

C# Copy

K is INTEGER

l  Int32

C# Copy

L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.

a  Span<TReal>

C# Copy

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

lda  Int32

C# Copy

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

b  Span<TReal>

C# Copy

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0.
See Purpose for details.

ldb  Int32

C# Copy

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

alpha  Span<TReal>

C# Copy

ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Span<TReal>

C# Copy

BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Span<TReal>

C# Copy

U is DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced.

ldu  Int32

C# Copy

LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Span<TReal>

C# Copy

V is DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced.

ldv  Int32

C# Copy

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Span<TReal>

C# Copy

Q is DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.

ldq  Int32

C# Copy

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Span<Int32>

C# Copy

IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(N).

info  Int32

C# Copy

INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine DTGSJA.

Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the orthogonal
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
also equal to the CS decomposition of A and B. Furthermore, the GSVD
can be used to derive the solution of the eigenvalue problem:
                     A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                     X = Q*( I   0    )
                           ( 0 inv(R) ).

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

DGGSVD3 replaces the deprecated subroutine DGGSVD.

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

Date: August 2015

public abstract void GeneralizedSingularValueDecompose(
	char jobu,
	char jobv,
	char jobq,
	int m,
	int n,
	int p,
	out int k,
	out int l,
	Span<TComplex> a,
	int lda,
	Span<TComplex> b,
	int ldb,
	Span<TReal> alpha,
	Span<TReal> beta,
	Span<TComplex> u,
	int ldu,
	Span<TComplex> v,
	int ldv,
	Span<TComplex> q,
	int ldq,
	Span<int> ipiv,
	out int info
)

Public MustOverride Sub GeneralizedSingularValueDecompose ( 
	jobu As Char,
	jobv As Char,
	jobq As Char,
	m As Integer,
	n As Integer,
	p As Integer,
	<OutAttribute> ByRef k As Integer,
	<OutAttribute> ByRef l As Integer,
	a As Span(Of TComplex),
	lda As Integer,
	b As Span(Of TComplex),
	ldb As Integer,
	alpha As Span(Of TReal),
	beta As Span(Of TReal),
	u As Span(Of TComplex),
	ldu As Integer,
	v As Span(Of TComplex),
	ldv As Integer,
	q As Span(Of TComplex),
	ldq As Integer,
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void GeneralizedSingularValueDecompose(
	wchar_t jobu, 
	wchar_t jobv, 
	wchar_t jobq, 
	int m, 
	int n, 
	int p, 
	[OutAttribute] int% k, 
	[OutAttribute] int% l, 
	Span<TComplex> a, 
	int lda, 
	Span<TComplex> b, 
	int ldb, 
	Span<TReal> alpha, 
	Span<TReal> beta, 
	Span<TComplex> u, 
	int ldu, 
	Span<TComplex> v, 
	int ldv, 
	Span<TComplex> q, 
	int ldq, 
	Span<int> ipiv, 
	[OutAttribute] int% info
) abstract

abstract GeneralizedSingularValueDecompose : 
        jobu : char * 
        jobv : char * 
        jobq : char * 
        m : int * 
        n : int * 
        p : int * 
        k : int byref * 
        l : int byref * 
        a : Span<'TComplex> * 
        lda : int * 
        b : Span<'TComplex> * 
        ldb : int * 
        alpha : Span<'TReal> * 
        beta : Span<'TReal> * 
        u : Span<'TComplex> * 
        ldu : int * 
        v : Span<'TComplex> * 
        ldv : int * 
        q : Span<'TComplex> * 
        ldq : int * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

jobu  Char

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JOBU is CHARACTER*1
= 'U':  Unitary matrix U is computed;
= 'N':  U is not computed.

jobv  Char

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JOBV is CHARACTER*1
= 'V':  Unitary matrix V is computed;
= 'N':  V is not computed.

jobq  Char

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JOBQ is CHARACTER*1
= 'Q':  Unitary matrix Q is computed;
= 'N':  Q is not computed.

m  Int32

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M is INTEGER
The number of rows of the matrix A.  M >= 0.

n  Int32

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N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

p  Int32

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P is INTEGER
The number of rows of the matrix B.  P >= 0.

k  Int32

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K is INTEGER

l  Int32

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L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**H,B**H)**H.

a  Span<TComplex>

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A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.

lda  Int32

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

b  Span<TComplex>

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B is COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of the triangular matrix R if
M-K-L < 0.  See Purpose for details.

ldb  Int32

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

alpha  Span<TReal>

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ALPHA is DOUBLE PRECISION array, dimension (N)

beta  Span<TReal>

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BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
  ALPHA(1:K) = 1,
  BETA(1:K)  = 0,
and if M-K-L >= 0,
  ALPHA(K+1:K+L) = C,
  BETA(K+1:K+L)  = S,
or if M-K-L < 0,
  ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
  BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
  ALPHA(K+L+1:N) = 0
  BETA(K+L+1:N)  = 0

u  Span<TComplex>

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U is COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary matrix U.
If JOBU = 'N', U is not referenced.

ldu  Int32

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LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

v  Span<TComplex>

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V is COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary matrix V.
If JOBV = 'N', V is not referenced.

ldv  Int32

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LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

q  Span<TComplex>

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Q is COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
If JOBQ = 'N', Q is not referenced.

ldq  Int32

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LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

ipiv  Span<Int32>

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IPIV is INTEGER array, dimension (N)
On exit, IPIV stores the sorting information. More
precisely, the following loop will sort ALPHA
   for I = K+1, min(M,K+L)
       swap ALPHA(I) and ALPHA(IPIV(I))
   endfor
such that ALPHA(1) >;= ALPHA(2) >= ... >= ALPHA(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.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
      converge.  For further details, see subroutine ZTGSJA.

Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
matrices and of the following structures, respectively:
If M-K-L >;= 0,
                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )
                  K  L
       D2 =   L ( 0  S )
            P-L ( 0  0 )
                N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 )
            L (  0    0   R22 )
where
  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.
  R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L <; 0,
                  K M-K K+L-M
       D1 =   K ( I  0    0   )
            M-K ( 0  C    0   )
                    K M-K K+L-M
       D2 =   M-K ( 0  S    0  )
            K+L-M ( 0  0    I  )
              P-L ( 0  0    0  )
                   N-K-L  K   M-K  K+L-M
  ( 0 R ) =     K ( 0    R11  R12  R13  )
              M-K ( 0     0   R22  R23  )
            K+L-M ( 0     0    0   R33  )
where
  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
  S = diag( BETA(K+1),  ... , BETA(M) ),
  C**2 + S**2 = I.
  (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
  ( 0  R22 R23 )
  in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the unitary
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
                     A*inv(B) = U*(D1*inv(D2))*V**H.
If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
equal to the CS decomposition of A and B. Furthermore, the GSVD can
be used to derive the solution of the eigenvalue problem:
                     A**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
                      X = Q*(  I   0    )
                            (  0 inv(R) )

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

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

Date: August 2015

Reference