org.netlib.lapack
Class Sggsvd
java.lang.Object
org.netlib.lapack.Sggsvd
public class Sggsvd
- 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
* =======
*
* SGGSVD computes the generalized singular value decomposition (GSVD)
* of an M-by-N real matrix A and P-by-N real matrix B:
*
* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
*
* where U, V and Q are orthogonal matrices, and Z' is the transpose
* of Z. Let K+L = the effective numerical rank of the matrix (A',B')',
* 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'.
* If ( A',B')' 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'*A x = lambda* B'*B x.
* In some literature, the GSVD of A and B is presented in the form
* U'*A*X = ( 0 D1 ), V'*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) ).
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* = 'U': Orthogonal matrix U is computed;
* = 'N': U is not computed.
*
* JOBV (input) CHARACTER*1
* = 'V': Orthogonal matrix V is computed;
* = 'N': V is not computed.
*
* JOBQ (input) CHARACTER*1
* = 'Q': Orthogonal matrix Q is computed;
* = 'N': Q is not computed.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* K (output) INTEGER
* L (output) INTEGER
* On exit, K and L specify the dimension of the subblocks
* described in the Purpose section.
* K + L = effective numerical rank of (A',B')'.
*
* A (input/output) REAL 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 (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) REAL 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 (input) INTEGER
* The leading dimension of the array B. LDA >= max(1,P).
*
* ALPHA (output) REAL array, dimension (N)
* BETA (output) REAL 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 (output) REAL array, dimension (LDU,M)
* If JOBU = 'U', U contains the M-by-M orthogonal matrix 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 (output) REAL array, dimension (LDV,P)
* If JOBV = 'V', V contains the P-by-P orthogonal matrix 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 (output) REAL array, dimension (LDQ,N)
* If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix 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) REAL array,
* dimension (max(3*N,M,P)+N)
*
* IWORK (workspace/output) INTEGER array, dimension (N)
* On exit, IWORK 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(IWORK(I))
* endfor
* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*
* INFO (output)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 STGSJA.
*
* Internal Parameters
* ===================
*
* TOLA REAL
* TOLB REAL
* TOLA and TOLB are the thresholds to determine the effective
* rank of (A',B')'. Generally, they are set to
* TOLA = MAX(M,N)*norm(A)*MACHEPS,
* TOLB = MAX(P,N)*norm(B)*MACHEPS.
* The size of TOLA and TOLB may affect the size of backward
* errors of the decomposition.
*
* Further Details
* ===============
*
* 2-96 Based on modifications by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Local Scalars ..
|
Method Summary |
static void |
sggsvd(java.lang.String jobu,
java.lang.String jobv,
java.lang.String jobq,
int m,
int n,
int p,
intW k,
intW l,
float[] a,
int _a_offset,
int lda,
float[] b,
int _b_offset,
int ldb,
float[] alpha,
int _alpha_offset,
float[] beta,
int _beta_offset,
float[] u,
int _u_offset,
int ldu,
float[] v,
int _v_offset,
int ldv,
float[] q,
int _q_offset,
int ldq,
float[] work,
int _work_offset,
int[] iwork,
int _iwork_offset,
intW info)
|
| Methods inherited from class java.lang.Object |
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Sggsvd
public Sggsvd()
sggsvd
public static void sggsvd(java.lang.String jobu,
java.lang.String jobv,
java.lang.String jobq,
int m,
int n,
int p,
intW k,
intW l,
float[] a,
int _a_offset,
int lda,
float[] b,
int _b_offset,
int ldb,
float[] alpha,
int _alpha_offset,
float[] beta,
int _beta_offset,
float[] u,
int _u_offset,
int ldu,
float[] v,
int _v_offset,
int ldv,
float[] q,
int _q_offset,
int ldq,
float[] work,
int _work_offset,
int[] iwork,
int _iwork_offset,
intW info)