Managed
Definition
Namespace: Numerics.NET.LinearAlgebra.Implementation
Assembly: Numerics.NET (in Numerics.NET.dll) Version: 9.0.7
Assembly: Numerics.NET (in Numerics.NET.dll) Version: 9.0.7
Overload List
GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Span<Complex<Double>>, Int32, Span<Complex<Double>>, Int32, Span<Double>, Span<Double>, Span<Complex<Double>>, Int32, Span<Complex<Double>>, Int32, Span<Complex<Double>>, Int32, Span<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.
public override void GeneralizedSingularValueDecompose(
char jobu,
char jobv,
char jobq,
int m,
int n,
int p,
out int k,
out int l,
Span<Complex<double>> a,
int lda,
Span<Complex<double>> b,
int ldb,
Span<double> alpha,
Span<double> beta,
Span<Complex<double>> u,
int ldu,
Span<Complex<double>> v,
int ldv,
Span<Complex<double>> q,
int ldq,
Span<int> ipiv,
out int info
)Parameters
- jobu Char
JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed.
- jobv Char
JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed.
- jobq Char
JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed.
- m Int32
M is INTEGER The number of rows of the matrix A. M >= 0.
- n Int32
N is INTEGER The number of columns of the matrices A and B. N >= 0.
- p Int32
P is INTEGER The number of rows of the matrix B. P >= 0.
- k Int32
K is INTEGER
- l Int32
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<Complex<Double>>
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
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
- b Span<Complex<Double>>
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
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).
- alpha Span<Double>
ALPHA is DOUBLE PRECISION array, dimension (N)
- beta Span<Double>
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<Complex<Double>>
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
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
- v Span<Complex<Double>>
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
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
- q Span<Complex<Double>>
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
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
- ipiv Span<Int32>
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
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
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
GeneralizedSingularValueDecompose(Char, Char, Char, Int32, Int32, Int32, Int32, Int32, Span<Double>, Int32, Span<Double>, Int32, Span<Double>, Span<Double>, Span<Double>, Int32, Span<Double>, Int32, Span<Double>, Int32, Span<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.
public override void GeneralizedSingularValueDecompose(
char jobu,
char jobv,
char jobq,
int m,
int n,
int p,
out int k,
out int l,
Span<double> a,
int lda,
Span<double> b,
int ldb,
Span<double> alpha,
Span<double> beta,
Span<double> u,
int ldu,
Span<double> v,
int ldv,
Span<double> q,
int ldq,
Span<int> ipiv,
out int info
)Parameters
- jobu Char
JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed.
- jobv Char
JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed.
- jobq Char
JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed.
- m Int32
M is INTEGER The number of rows of the matrix A. M >= 0.
- n Int32
N is INTEGER The number of columns of the matrices A and B. N >= 0.
- p Int32
P is INTEGER The number of rows of the matrix B. P >= 0.
- k Int32
K is INTEGER
- l Int32
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<Double>
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
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
- b Span<Double>
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
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).
- alpha Span<Double>
ALPHA is DOUBLE PRECISION array, dimension (N)
- beta Span<Double>
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<Double>
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
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
- v Span<Double>
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
LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
- q Span<Double>
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
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
- ipiv Span<Int32>
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
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
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