DecompositionOperations<TReal, TComplex>.GeneralizedSingularValueDecompose Method

Definition

Namespace: Extreme.Mathematics.LinearAlgebra.Implementation
Assembly: Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.23

Overload List

GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Array2D<TReal>, Array2D<TReal>, Array1D<TReal>, Array1D<TReal>, Array2D<TReal>, Array2D<TReal>, Array2D<TReal>, Array1D<Int32>, Int32)

Computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices.

GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Array2D<TComplex>, Array2D<TComplex>, Array1D<TReal>, Array1D<TReal>, Array2D<TComplex>, Array2D<TComplex>, Array2D<TComplex>, Array1D<Int32>, Int32)

Computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B: U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R ) where U, V and Q are unitary matrices.

GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Array2D<TReal>, Array2D<TReal>, Array1D<TReal>, Array1D<TReal>, Array2D<TReal>, Array2D<TReal>, Array2D<TReal>, Array1D<Int32>, Int32)

Computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices.

C# 
public abstract 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
)

Parameters

jobu  Char
C# 
JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.
jobv  Char
C# 
JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.
jobq  Char
C# 
JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.
m  Int32
C# 
M is INTEGER
The number of rows of the matrix A.  M >= 0.
n  Int32
C# 
N is INTEGER
The number of columns of the matrices A and B.  N >= 0.
p  Int32
C# 
P is INTEGER
The number of rows of the matrix B.  P >= 0.
k  Int32
C# 
K is INTEGER
l  Int32
C# 
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>
C# 
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.
b  Array2D<TReal>
C# 
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.
alpha  Array1D<TReal>
C# 
ALPHA is DOUBLE PRECISION array, dimension (N)
beta  Array1D<TReal>
C# 
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>
C# 
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.
v  Array2D<TReal>
C# 
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.
q  Array2D<TReal>
C# 
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.
ipiv  Array1D<Int32>
C# 
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# 
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.

Remarks

C# 
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

GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Array2D<TComplex>, Array2D<TComplex>, Array1D<TReal>, Array1D<TReal>, Array2D<TComplex>, Array2D<TComplex>, Array2D<TComplex>, Array1D<Int32>, Int32)

Computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B: U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R ) where U, V and Q are unitary matrices.

C# 
public abstract 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
)

Parameters

jobu  Char
C# 
JOBU is CHARACTER*1
= 'U':  Unitary matrix U is computed;
= 'N':  U is not computed.
jobv  Char
C# 
JOBV is CHARACTER*1
= 'V':  Unitary matrix V is computed;
= 'N':  V is not computed.
jobq  Char
C# 
JOBQ is CHARACTER*1
= 'Q':  Unitary matrix Q is computed;
= 'N':  Q is not computed.
m  Int32
C# 
M is INTEGER
The number of rows of the matrix A.  M >= 0.
n  Int32
C# 
N is INTEGER
The number of columns of the matrices A and B.  N >= 0.
p  Int32
C# 
P is INTEGER
The number of rows of the matrix B.  P >= 0.
k  Int32
C# 
K is INTEGER
l  Int32
C# 
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>
C# 
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.
b  Array2D<TComplex>
C# 
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.
alpha  Array1D<TReal>
C# 
ALPHA is DOUBLE PRECISION array, dimension (N)
beta  Array1D<TReal>
C# 
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>
C# 
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.
v  Array2D<TComplex>
C# 
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.
q  Array2D<TComplex>
C# 
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.
ipiv  Array1D<Int32>
C# 
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# 
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.

Remarks

C# 
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

See Also