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SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
$ X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, $ TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO ) IMPLICIT NONE * * -- LAPACK routine (version 3.3.0) -- * * -- Contributed by Brian Sutton of the Randolph-Macon College -- * -- November 2010 * * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER SIGNS, TRANS INTEGER INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P, $ Q * .. * .. Array Arguments .. DOUBLE PRECISION PHI( * ), THETA( * ) DOUBLE PRECISION TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ), $ WORK( * ), X11( LDX11, * ), X12( LDX12, * ), $ X21( LDX21, * ), X22( LDX22, * ) * .. * * Purpose * ======= * * DORBDB simultaneously bidiagonalizes the blocks of an M-by-M * partitioned orthogonal matrix X: * * [ B11 | B12 0 0 ] * [ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T * X = [-----------] = [---------] [----------------] [---------] . * [ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ] * [ 0 | 0 0 I ] * * X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is * not the case, then X must be transposed and/or permuted. This can be * done in constant time using the TRANS and SIGNS options. See DORCSD * for details.) * * The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by- * (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are * represented implicitly by Householder vectors. * * B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented * implicitly by angles THETA, PHI. * * Arguments * ========= * * 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 <= * MIN(P,M-P,M-Q). * * X11 (input/output) DOUBLE PRECISION array, dimension (LDX11,Q) * On entry, the top-left block of the orthogonal matrix to be * reduced. On exit, the form depends on TRANS: * If TRANS = 'N', then * the columns of tril(X11) specify reflectors for P1, * the rows of triu(X11,1) specify reflectors for Q1; * else TRANS = 'T', and * the rows of triu(X11) specify reflectors for P1, * the columns of tril(X11,-1) specify reflectors for Q1. * * LDX11 (input) INTEGER * The leading dimension of X11. If TRANS = 'N', then LDX11 >= * P; else LDX11 >= Q. * * X12 (input/output) DOUBLE PRECISION array, dimension (LDX12,M-Q) * On entry, the top-right block of the orthogonal matrix to * be reduced. On exit, the form depends on TRANS: * If TRANS = 'N', then * the rows of triu(X12) specify the first P reflectors for * Q2; * else TRANS = 'T', and * the columns of tril(X12) specify the first P reflectors * for Q2. * * LDX12 (input) INTEGER * The leading dimension of X12. If TRANS = 'N', then LDX12 >= * P; else LDX11 >= M-Q. * * X21 (input/output) DOUBLE PRECISION array, dimension (LDX21,Q) * On entry, the bottom-left block of the orthogonal matrix to * be reduced. On exit, the form depends on TRANS: * If TRANS = 'N', then * the columns of tril(X21) specify reflectors for P2; * else TRANS = 'T', and * the rows of triu(X21) specify reflectors for P2. * * LDX21 (input) INTEGER * The leading dimension of X21. If TRANS = 'N', then LDX21 >= * M-P; else LDX21 >= Q. * * X22 (input/output) DOUBLE PRECISION array, dimension (LDX22,M-Q) * On entry, the bottom-right block of the orthogonal matrix to * be reduced. On exit, the form depends on TRANS: * If TRANS = 'N', then * the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last * M-P-Q reflectors for Q2, * else TRANS = 'T', and * the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last * M-P-Q reflectors for P2. * * LDX22 (input) INTEGER * The leading dimension of X22. If TRANS = 'N', then LDX22 >= * M-P; else LDX22 >= M-Q. * * THETA (output) DOUBLE PRECISION array, dimension (Q) * The entries of the bidiagonal blocks B11, B12, B21, B22 can * be computed from the angles THETA and PHI. See Further * Details. * * PHI (output) DOUBLE PRECISION array, dimension (Q-1) * The entries of the bidiagonal blocks B11, B12, B21, B22 can * be computed from the angles THETA and PHI. See Further * Details. * * TAUP1 (output) DOUBLE PRECISION array, dimension (P) * The scalar factors of the elementary reflectors that define * P1. * * TAUP2 (output) DOUBLE PRECISION array, dimension (M-P) * The scalar factors of the elementary reflectors that define * P2. * * TAUQ1 (output) DOUBLE PRECISION array, dimension (Q) * The scalar factors of the elementary reflectors that define * Q1. * * TAUQ2 (output) DOUBLE PRECISION array, dimension (M-Q) * The scalar factors of the elementary reflectors that define * Q2. * * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK) * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= M-Q. * * 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. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * The bidiagonal blocks B11, B12, B21, and B22 are represented * implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., * PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are * lower bidiagonal. Every entry in each bidiagonal band is a product * of a sine or cosine of a THETA with a sine or cosine of a PHI. See * [1] or DORCSD for details. * * P1, P2, Q1, and Q2 are represented as products of elementary * reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2 * using DORGQR and DORGLQ. * * Reference * ========= * * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. * Algorithms, 50(1):33-65, 2009. * * ==================================================================== * * .. Parameters .. DOUBLE PRECISION REALONE PARAMETER ( REALONE = 1.0D0 ) DOUBLE PRECISION NEGONE, ONE PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL COLMAJOR, LQUERY INTEGER I, LWORKMIN, LWORKOPT DOUBLE PRECISION Z1, Z2, Z3, Z4 * .. * .. External Subroutines .. EXTERNAL DAXPY, DLARF, DLARFGP, DSCAL, XERBLA * .. * .. External Functions .. DOUBLE PRECISION DNRM2 LOGICAL LSAME EXTERNAL DNRM2, LSAME * .. * .. Intrinsic Functions INTRINSIC ATAN2, COS, MAX, MIN, SIN * .. * .. Executable Statements .. * * Test input arguments * INFO = 0 COLMAJOR = .NOT. LSAME( TRANS, 'T' ) IF( .NOT. LSAME( SIGNS, 'O' ) ) THEN Z1 = REALONE Z2 = REALONE Z3 = REALONE Z4 = REALONE ELSE Z1 = REALONE Z2 = -REALONE Z3 = REALONE Z4 = -REALONE END IF LQUERY = LWORK .EQ. -1 * IF( M .LT. 0 ) THEN INFO = -3 ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN INFO = -4 ELSE IF( Q .LT. 0 .OR. Q .GT. P .OR. Q .GT. M-P .OR. $ Q .GT. M-Q ) THEN INFO = -5 ELSE IF( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN INFO = -7 ELSE IF( .NOT.COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN INFO = -7 ELSE IF( COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN INFO = -9 ELSE IF( .NOT.COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN INFO = -9 ELSE IF( COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN INFO = -11 ELSE IF( .NOT.COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN INFO = -11 ELSE IF( COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN INFO = -13 ELSE IF( .NOT.COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN INFO = -13 END IF * * Compute workspace * IF( INFO .EQ. 0 ) THEN LWORKOPT = M - Q LWORKMIN = M - Q WORK(1) = LWORKOPT IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN INFO = -21 END IF END IF IF( INFO .NE. 0 ) THEN CALL XERBLA( 'xORBDB', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Handle column-major and row-major separately * IF( COLMAJOR ) THEN * * Reduce columns 1, ..., Q of X11, X12, X21, and X22 * DO I = 1, Q * IF( I .EQ. 1 ) THEN CALL DSCAL( P-I+1, Z1, X11(I,I), 1 ) ELSE CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), 1 ) CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I,I-1), $ 1, X11(I,I), 1 ) END IF IF( I .EQ. 1 ) THEN CALL DSCAL( M-P-I+1, Z2, X21(I,I), 1 ) ELSE CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), 1 ) CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I,I-1), $ 1, X21(I,I), 1 ) END IF * THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), 1 ), $ DNRM2( P-I+1, X11(I,I), 1 ) ) * CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) ) X11(I,I) = ONE CALL DLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) ) X21(I,I) = ONE * CALL DLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), $ X11(I,I+1), LDX11, WORK ) CALL DLARF( 'L', P-I+1, M-Q-I+1, X11(I,I), 1, TAUP1(I), $ X12(I,I), LDX12, WORK ) CALL DLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, TAUP2(I), $ X21(I,I+1), LDX21, WORK ) CALL DLARF( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1, TAUP2(I), $ X22(I,I), LDX22, WORK ) * IF( I .LT. Q ) THEN CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I,I+1), $ LDX11 ) CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I,I+1), LDX21, $ X11(I,I+1), LDX11 ) END IF CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), LDX12 ) CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), LDX22, $ X12(I,I), LDX12 ) * IF( I .LT. Q ) $ PHI(I) = ATAN2( DNRM2( Q-I, X11(I,I+1), LDX11 ), $ DNRM2( M-Q-I+1, X12(I,I), LDX12 ) ) * IF( I .LT. Q ) THEN CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11, $ TAUQ1(I) ) X11(I,I+1) = ONE END IF CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12, $ TAUQ2(I) ) X12(I,I) = ONE * IF( I .LT. Q ) THEN CALL DLARF( 'R', P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I), $ X11(I+1,I+1), LDX11, WORK ) CALL DLARF( 'R', M-P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I), $ X21(I+1,I+1), LDX21, WORK ) END IF CALL DLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I), $ X12(I+1,I), LDX12, WORK ) CALL DLARF( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I), $ X22(I+1,I), LDX22, WORK ) * END DO * * Reduce columns Q + 1, ..., P of X12, X22 * DO I = Q + 1, P * CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), LDX12 ) CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12, $ TAUQ2(I) ) X12(I,I) = ONE * CALL DLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I), $ X12(I+1,I), LDX12, WORK ) IF( M-P-Q .GE. 1 ) $ CALL DLARF( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12, $ TAUQ2(I), X22(Q+1,I), LDX22, WORK ) * END DO * * Reduce columns P + 1, ..., M - Q of X12, X22 * DO I = 1, M - P - Q * CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(Q+I,P+I), LDX22 ) CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I+1), $ LDX22, TAUQ2(P+I) ) X22(Q+I,P+I) = ONE CALL DLARF( 'R', M-P-Q-I, M-P-Q-I+1, X22(Q+I,P+I), LDX22, $ TAUQ2(P+I), X22(Q+I+1,P+I), LDX22, WORK ) * END DO * ELSE * * Reduce columns 1, ..., Q of X11, X12, X21, X22 * DO I = 1, Q * IF( I .EQ. 1 ) THEN CALL DSCAL( P-I+1, Z1, X11(I,I), LDX11 ) ELSE CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), LDX11 ) CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I-1,I), $ LDX12, X11(I,I), LDX11 ) END IF IF( I .EQ. 1 ) THEN CALL DSCAL( M-P-I+1, Z2, X21(I,I), LDX21 ) ELSE CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), LDX21 ) CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I-1,I), $ LDX22, X21(I,I), LDX21 ) END IF * THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), LDX21 ), $ DNRM2( P-I+1, X11(I,I), LDX11 ) ) * CALL DLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11, TAUP1(I) ) X11(I,I) = ONE CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21, $ TAUP2(I) ) X21(I,I) = ONE * CALL DLARF( 'R', Q-I, P-I+1, X11(I,I), LDX11, TAUP1(I), $ X11(I+1,I), LDX11, WORK ) CALL DLARF( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11, TAUP1(I), $ X12(I,I), LDX12, WORK ) CALL DLARF( 'R', Q-I, M-P-I+1, X21(I,I), LDX21, TAUP2(I), $ X21(I+1,I), LDX21, WORK ) CALL DLARF( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21, $ TAUP2(I), X22(I,I), LDX22, WORK ) * IF( I .LT. Q ) THEN CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I+1,I), 1 ) CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I+1,I), 1, $ X11(I+1,I), 1 ) END IF CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), 1 ) CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), 1, $ X12(I,I), 1 ) * IF( I .LT. Q ) $ PHI(I) = ATAN2( DNRM2( Q-I, X11(I+1,I), 1 ), $ DNRM2( M-Q-I+1, X12(I,I), 1 ) ) * IF( I .LT. Q ) THEN CALL DLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1, TAUQ1(I) ) X11(I+1,I) = ONE END IF CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) ) X12(I,I) = ONE * IF( I .LT. Q ) THEN CALL DLARF( 'L', Q-I, P-I, X11(I+1,I), 1, TAUQ1(I), $ X11(I+1,I+1), LDX11, WORK ) CALL DLARF( 'L', Q-I, M-P-I, X11(I+1,I), 1, TAUQ1(I), $ X21(I+1,I+1), LDX21, WORK ) END IF CALL DLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I), $ X12(I,I+1), LDX12, WORK ) CALL DLARF( 'L', M-Q-I+1, M-P-I, X12(I,I), 1, TAUQ2(I), $ X22(I,I+1), LDX22, WORK ) * END DO * * Reduce columns Q + 1, ..., P of X12, X22 * DO I = Q + 1, P * CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), 1 ) CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) ) X12(I,I) = ONE * CALL DLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I), $ X12(I,I+1), LDX12, WORK ) IF( M-P-Q .GE. 1 ) $ CALL DLARF( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1, TAUQ2(I), $ X22(I,Q+1), LDX22, WORK ) * END DO * * Reduce columns P + 1, ..., M - Q of X12, X22 * DO I = 1, M - P - Q * CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(P+I,Q+I), 1 ) CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I+1,Q+I), 1, $ TAUQ2(P+I) ) X22(P+I,Q+I) = ONE * CALL DLARF( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I), 1, $ TAUQ2(P+I), X22(P+I,Q+I+1), LDX22, WORK ) * END DO * END IF * RETURN * * End of DORBDB * END |