org.netlib.lapack
Class Dtgsja
java.lang.Object
org.netlib.lapack.Dtgsja
public class Dtgsja
- extends java.lang.Object
Following is the description from the original
Fortran source. For each array argument, the Java
version will include an integer offset parameter, so
the arguments may not match the description exactly.
Contact seymour@cs.utk.edu with any questions.
* ..
*
* Purpose
* =======
*
* 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'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
*
* where U, V and Q are orthogonal matrices, Z' denotes the transpose
* of Z, 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.
*
* Arguments
* =========
*
* JOBU (input) 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.
*
* JOBV (input) 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.
*
* JOBQ (input) 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.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* K (input) INTEGER
* L (input) 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.
*
* A (input/output) 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) 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.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* TOLA (input) DOUBLE PRECISION
* TOLB (input) 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.
*
* ALPHA (output) DOUBLE PRECISION array, dimension (N)
* BETA (output) 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.
*
* U (input/output) 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.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= max(1,M) if
* JOBU = 'U'; LDU >= 1 otherwise.
*
* V (input/output) 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.
*
* LDV (input) INTEGER
* The leading dimension of the array V. LDV >= max(1,P) if
* JOBV = 'V'; LDV >= 1 otherwise.
*
* Q (input/output) 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.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,N) if
* JOBQ = 'Q'; LDQ >= 1 otherwise.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
*
* NCYCLE (output) INTEGER
* The number of cycles required for convergence.
*
* INFO (output) 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.
*
* Further Details
* ===============
*
* 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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
*
* where U1, V1 and Q1 are orthogonal matrix, and Z' 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.
*
* =====================================================================
*
* .. Parameters ..
|
Method Summary |
static void |
dtgsja(java.lang.String jobu,
java.lang.String jobv,
java.lang.String jobq,
int m,
int p,
int n,
int k,
int l,
double[] a,
int _a_offset,
int lda,
double[] b,
int _b_offset,
int ldb,
double tola,
double tolb,
double[] alpha,
int _alpha_offset,
double[] beta,
int _beta_offset,
double[] u,
int _u_offset,
int ldu,
double[] v,
int _v_offset,
int ldv,
double[] q,
int _q_offset,
int ldq,
double[] work,
int _work_offset,
intW ncycle,
intW info)
|
| Methods inherited from class java.lang.Object |
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Dtgsja
public Dtgsja()
dtgsja
public static void dtgsja(java.lang.String jobu,
java.lang.String jobv,
java.lang.String jobq,
int m,
int p,
int n,
int k,
int l,
double[] a,
int _a_offset,
int lda,
double[] b,
int _b_offset,
int ldb,
double tola,
double tolb,
double[] alpha,
int _alpha_offset,
double[] beta,
int _beta_offset,
double[] u,
int _u_offset,
int ldu,
double[] v,
int _v_offset,
int ldv,
double[] q,
int _q_offset,
int ldq,
double[] work,
int _work_offset,
intW ncycle,
intW info)