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LAPACK
3.6.1
LAPACK: Linear Algebra PACKage
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LAPACK
3.6.1
LAPACK: Linear Algebra PACKage
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| subroutine dtgsja | ( | character | JOBU, |
| character | JOBV, | ||
| character | JOBQ, | ||
| integer | M, | ||
| integer | P, | ||
| integer | N, | ||
| integer | K, | ||
| integer | L, | ||
| double precision, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| double precision, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| double precision | TOLA, | ||
| double precision | TOLB, | ||
| double precision, dimension( * ) | ALPHA, | ||
| double precision, dimension( * ) | BETA, | ||
| double precision, dimension( ldu, * ) | U, | ||
| integer | LDU, | ||
| double precision, dimension( ldv, * ) | V, | ||
| integer | LDV, | ||
| double precision, dimension( ldq, * ) | Q, | ||
| integer | LDQ, | ||
| double precision, dimension( * ) | WORK, | ||
| integer | NCYCLE, | ||
| integer | INFO | ||
| ) |
DTGSJA
Download DTGSJA + dependencies [TGZ] [ZIP] [TXT]
DTGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.
On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine DGGSVP
from a general M-by-N matrix A and P-by-N matrix B:
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.
On exit,
U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
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 ) K
L ( 0 0 R22 ) L
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.
R = ( 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 computation of the orthogonal transformation matrices U, V or Q
is optional. These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1. | [in] | JOBU | JOBU is CHARACTER*1
= 'U': U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
= 'I': U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= 'N': U is not computed. |
| [in] | JOBV | JOBV is CHARACTER*1
= 'V': V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
= 'I': V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= 'N': V is not computed. |
| [in] | JOBQ | JOBQ is CHARACTER*1
= 'Q': Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'N': Q is not computed. |
| [in] | M | M is INTEGER
The number of rows of the matrix A. M >= 0. |
| [in] | P | P is INTEGER
The number of rows of the matrix B. P >= 0. |
| [in] | N | N is INTEGER
The number of columns of the matrices A and B. N >= 0. |
| [in] | K | K is INTEGER |
| [in] | L | L is INTEGER
K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by DTGSJA.
See Further Details. |
| [in,out] | A | A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
matrix R or part of R. See Purpose for details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [in,out] | B | B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
a part of R. See Purpose for details. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P). |
| [in] | TOLA | TOLA is DOUBLE PRECISION |
| [in] | TOLB | TOLB is DOUBLE PRECISION
TOLA and TOLB are the convergence criteria for the Jacobi-
Kogbetliantz iteration procedure. Generally, they are the
same as used in the preprocessing step, say
TOLA = max(M,N)*norm(A)*MAZHEPS,
TOLB = max(P,N)*norm(B)*MAZHEPS. |
| [out] | ALPHA | ALPHA is DOUBLE PRECISION array, dimension (N) |
| [out] | BETA | 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) = diag(C),
BETA(K+1:K+L) = diag(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.
Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N) = 0. |
| [in,out] | U | U is DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBU = 'I', U contains the orthogonal matrix U;
if JOBU = 'U', U contains the product U1*U.
If JOBU = 'N', U is not referenced. |
| [in] | LDU | LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise. |
| [in,out] | V | V is DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBV = 'I', V contains the orthogonal matrix V;
if JOBV = 'V', V contains the product V1*V.
If JOBV = 'N', V is not referenced. |
| [in] | LDV | LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise. |
| [in,out] | Q | Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBQ = 'I', Q contains the orthogonal matrix Q;
if JOBQ = 'Q', Q contains the product Q1*Q.
If JOBQ = 'N', Q is not referenced. |
| [in] | LDQ | LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise. |
| [out] | WORK | WORK is DOUBLE PRECISION array, dimension (2*N) |
| [out] | NCYCLE | NCYCLE is INTEGER
The number of cycles required for convergence. |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles. |
Internal Parameters
===================
MAXIT INTEGER
MAXIT specifies the total loops that the iterative procedure
may take. If after MAXIT cycles, the routine fails to
converge, we return INFO = 1.
DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
matrix B13 to the form:
U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix. Definition at line 380 of file dtgsja.f.
1.8.10