Provided by: liblapack-doc_3.3.1-1_all

**NAME**

LAPACK-3 - computes the CS decomposition of an M-by-M partitioned orthogonal matrix X

**SYNOPSIS**

RECURSIVE SUBROUTINE SORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, IWORK, INFO ) IMPLICIT NONE CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12, LDX21, LDX22, LWORK, M, P, Q INTEGER IWORK( * ) REAL THETA( * ) REAL U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ), X12( LDX12, * ), X21( LDX21, * ), X22( LDX22, * )

**PURPOSE**

SORCSD computes the CS decomposition of an M-by-M partitioned orthogonal matrix X: [ I 0 0 | 0 0 0 ] [ 0 C 0 | 0 -S 0 ] [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T X = [-----------] = [---------] [---------------------] [---------] . [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ] [ 0 S 0 | 0 C 0 ] [ 0 0 I | 0 0 0 ] X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P, (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q).

**ARGUMENTS**

JOBU1 (input) CHARACTER = 'Y': U1 is computed; otherwise: U1 is not computed. JOBU2 (input) CHARACTER = 'Y': U2 is computed; otherwise: U2 is not computed. JOBV1T (input) CHARACTER = 'Y': V1T is computed; otherwise: V1T is not computed. JOBV2T (input) CHARACTER = 'Y': V2T is computed; otherwise: V2T is not computed. TRANS (input) CHARACTER = 'T': X, U1, U2, V1T, and V2T are stored in row-major order; otherwise: X, U1, U2, V1T, and V2T are stored in column- major order. SIGNS (input) CHARACTER = 'O': The lower-left block is made nonpositive (the "other" convention); otherwise: The upper-right block is made nonpositive (the "default" convention). M (input) INTEGER The number of rows and columns in X. P (input) INTEGER The number of rows in X11 and X12. 0 <= P <= M. Q (input) INTEGER The number of columns in X11 and X21. 0 <= Q <= M. X (input/workspace) REAL array, dimension (LDX,M) On entry, the orthogonal matrix whose CSD is desired. LDX (input) INTEGER The leading dimension of X. LDX >= MAX(1,M). THETA (output) REAL array, dimension (R), in which R = MIN(P,M-P,Q,M-Q). C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). U1 (output) REAL array, dimension (P) If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1. LDU1 (input) INTEGER The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P). U2 (output) REAL array, dimension (M-P) If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal matrix U2. LDU2 (input) INTEGER The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,M-P). V1T (output) REAL array, dimension (Q) If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal matrix V1**T. LDV1T (input) INTEGER The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q). V2T (output) REAL array, dimension (M-Q) If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T. LDV2T (input) INTEGER The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,M-Q). WORK (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. If INFO > 0 on exit, WORK(2:R) contains the values PHI(1), ..., PHI(R-1) that, together with THETA(1), ..., THETA(R), define the matrix in intermediate bidiagonal-block form remaining after nonconvergence. INFO specifies the number of nonzero PHI's. LWORK (input) INTEGER The dimension of the array WORK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the work array, and no error message related to LWORK is issued by XERBLA. IWORK (workspace) INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)) INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: SBBCSD did not converge. See the description of WORK above for details. Reference ========= [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009. LAPACK routine (version 3.3.1) April 2011 SORCSD(3lapack)