Annotation of rpl/lapack/lapack/zggsvd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
! 2: $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
! 3: $ RWORK, IWORK, INFO )
! 4: *
! 5: * -- LAPACK driver routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER JOBQ, JOBU, JOBV
! 12: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
! 13: * ..
! 14: * .. Array Arguments ..
! 15: INTEGER IWORK( * )
! 16: DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
! 17: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 18: $ U( LDU, * ), V( LDV, * ), WORK( * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * ZGGSVD computes the generalized singular value decomposition (GSVD)
! 25: * of an M-by-N complex matrix A and P-by-N complex matrix B:
! 26: *
! 27: * U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
! 28: *
! 29: * where U, V and Q are unitary matrices, and Z' means the conjugate
! 30: * transpose of Z. Let K+L = the effective numerical rank of the
! 31: * matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper
! 32: * triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
! 33: * matrices and of the following structures, respectively:
! 34: *
! 35: * If M-K-L >= 0,
! 36: *
! 37: * K L
! 38: * D1 = K ( I 0 )
! 39: * L ( 0 C )
! 40: * M-K-L ( 0 0 )
! 41: *
! 42: * K L
! 43: * D2 = L ( 0 S )
! 44: * P-L ( 0 0 )
! 45: *
! 46: * N-K-L K L
! 47: * ( 0 R ) = K ( 0 R11 R12 )
! 48: * L ( 0 0 R22 )
! 49: * where
! 50: *
! 51: * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
! 52: * S = diag( BETA(K+1), ... , BETA(K+L) ),
! 53: * C**2 + S**2 = I.
! 54: *
! 55: * R is stored in A(1:K+L,N-K-L+1:N) on exit.
! 56: *
! 57: * If M-K-L < 0,
! 58: *
! 59: * K M-K K+L-M
! 60: * D1 = K ( I 0 0 )
! 61: * M-K ( 0 C 0 )
! 62: *
! 63: * K M-K K+L-M
! 64: * D2 = M-K ( 0 S 0 )
! 65: * K+L-M ( 0 0 I )
! 66: * P-L ( 0 0 0 )
! 67: *
! 68: * N-K-L K M-K K+L-M
! 69: * ( 0 R ) = K ( 0 R11 R12 R13 )
! 70: * M-K ( 0 0 R22 R23 )
! 71: * K+L-M ( 0 0 0 R33 )
! 72: *
! 73: * where
! 74: *
! 75: * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
! 76: * S = diag( BETA(K+1), ... , BETA(M) ),
! 77: * C**2 + S**2 = I.
! 78: *
! 79: * (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
! 80: * ( 0 R22 R23 )
! 81: * in B(M-K+1:L,N+M-K-L+1:N) on exit.
! 82: *
! 83: * The routine computes C, S, R, and optionally the unitary
! 84: * transformation matrices U, V and Q.
! 85: *
! 86: * In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
! 87: * A and B implicitly gives the SVD of A*inv(B):
! 88: * A*inv(B) = U*(D1*inv(D2))*V'.
! 89: * If ( A',B')' has orthnormal columns, then the GSVD of A and B is also
! 90: * equal to the CS decomposition of A and B. Furthermore, the GSVD can
! 91: * be used to derive the solution of the eigenvalue problem:
! 92: * A'*A x = lambda* B'*B x.
! 93: * In some literature, the GSVD of A and B is presented in the form
! 94: * U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 )
! 95: * where U and V are orthogonal and X is nonsingular, and D1 and D2 are
! 96: * ``diagonal''. The former GSVD form can be converted to the latter
! 97: * form by taking the nonsingular matrix X as
! 98: *
! 99: * X = Q*( I 0 )
! 100: * ( 0 inv(R) )
! 101: *
! 102: * Arguments
! 103: * =========
! 104: *
! 105: * JOBU (input) CHARACTER*1
! 106: * = 'U': Unitary matrix U is computed;
! 107: * = 'N': U is not computed.
! 108: *
! 109: * JOBV (input) CHARACTER*1
! 110: * = 'V': Unitary matrix V is computed;
! 111: * = 'N': V is not computed.
! 112: *
! 113: * JOBQ (input) CHARACTER*1
! 114: * = 'Q': Unitary matrix Q is computed;
! 115: * = 'N': Q is not computed.
! 116: *
! 117: * M (input) INTEGER
! 118: * The number of rows of the matrix A. M >= 0.
! 119: *
! 120: * N (input) INTEGER
! 121: * The number of columns of the matrices A and B. N >= 0.
! 122: *
! 123: * P (input) INTEGER
! 124: * The number of rows of the matrix B. P >= 0.
! 125: *
! 126: * K (output) INTEGER
! 127: * L (output) INTEGER
! 128: * On exit, K and L specify the dimension of the subblocks
! 129: * described in Purpose.
! 130: * K + L = effective numerical rank of (A',B')'.
! 131: *
! 132: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 133: * On entry, the M-by-N matrix A.
! 134: * On exit, A contains the triangular matrix R, or part of R.
! 135: * See Purpose for details.
! 136: *
! 137: * LDA (input) INTEGER
! 138: * The leading dimension of the array A. LDA >= max(1,M).
! 139: *
! 140: * B (input/output) COMPLEX*16 array, dimension (LDB,N)
! 141: * On entry, the P-by-N matrix B.
! 142: * On exit, B contains part of the triangular matrix R if
! 143: * M-K-L < 0. See Purpose for details.
! 144: *
! 145: * LDB (input) INTEGER
! 146: * The leading dimension of the array B. LDB >= max(1,P).
! 147: *
! 148: * ALPHA (output) DOUBLE PRECISION array, dimension (N)
! 149: * BETA (output) DOUBLE PRECISION array, dimension (N)
! 150: * On exit, ALPHA and BETA contain the generalized singular
! 151: * value pairs of A and B;
! 152: * ALPHA(1:K) = 1,
! 153: * BETA(1:K) = 0,
! 154: * and if M-K-L >= 0,
! 155: * ALPHA(K+1:K+L) = C,
! 156: * BETA(K+1:K+L) = S,
! 157: * or if M-K-L < 0,
! 158: * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
! 159: * BETA(K+1:M) = S, BETA(M+1:K+L) = 1
! 160: * and
! 161: * ALPHA(K+L+1:N) = 0
! 162: * BETA(K+L+1:N) = 0
! 163: *
! 164: * U (output) COMPLEX*16 array, dimension (LDU,M)
! 165: * If JOBU = 'U', U contains the M-by-M unitary matrix U.
! 166: * If JOBU = 'N', U is not referenced.
! 167: *
! 168: * LDU (input) INTEGER
! 169: * The leading dimension of the array U. LDU >= max(1,M) if
! 170: * JOBU = 'U'; LDU >= 1 otherwise.
! 171: *
! 172: * V (output) COMPLEX*16 array, dimension (LDV,P)
! 173: * If JOBV = 'V', V contains the P-by-P unitary matrix V.
! 174: * If JOBV = 'N', V is not referenced.
! 175: *
! 176: * LDV (input) INTEGER
! 177: * The leading dimension of the array V. LDV >= max(1,P) if
! 178: * JOBV = 'V'; LDV >= 1 otherwise.
! 179: *
! 180: * Q (output) COMPLEX*16 array, dimension (LDQ,N)
! 181: * If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
! 182: * If JOBQ = 'N', Q is not referenced.
! 183: *
! 184: * LDQ (input) INTEGER
! 185: * The leading dimension of the array Q. LDQ >= max(1,N) if
! 186: * JOBQ = 'Q'; LDQ >= 1 otherwise.
! 187: *
! 188: * WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
! 189: *
! 190: * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
! 191: *
! 192: * IWORK (workspace/output) INTEGER array, dimension (N)
! 193: * On exit, IWORK stores the sorting information. More
! 194: * precisely, the following loop will sort ALPHA
! 195: * for I = K+1, min(M,K+L)
! 196: * swap ALPHA(I) and ALPHA(IWORK(I))
! 197: * endfor
! 198: * such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
! 199: *
! 200: * INFO (output) INTEGER
! 201: * = 0: successful exit.
! 202: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 203: * > 0: if INFO = 1, the Jacobi-type procedure failed to
! 204: * converge. For further details, see subroutine ZTGSJA.
! 205: *
! 206: * Internal Parameters
! 207: * ===================
! 208: *
! 209: * TOLA DOUBLE PRECISION
! 210: * TOLB DOUBLE PRECISION
! 211: * TOLA and TOLB are the thresholds to determine the effective
! 212: * rank of (A',B')'. Generally, they are set to
! 213: * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
! 214: * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
! 215: * The size of TOLA and TOLB may affect the size of backward
! 216: * errors of the decomposition.
! 217: *
! 218: * Further Details
! 219: * ===============
! 220: *
! 221: * 2-96 Based on modifications by
! 222: * Ming Gu and Huan Ren, Computer Science Division, University of
! 223: * California at Berkeley, USA
! 224: *
! 225: * =====================================================================
! 226: *
! 227: * .. Local Scalars ..
! 228: LOGICAL WANTQ, WANTU, WANTV
! 229: INTEGER I, IBND, ISUB, J, NCYCLE
! 230: DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
! 231: * ..
! 232: * .. External Functions ..
! 233: LOGICAL LSAME
! 234: DOUBLE PRECISION DLAMCH, ZLANGE
! 235: EXTERNAL LSAME, DLAMCH, ZLANGE
! 236: * ..
! 237: * .. External Subroutines ..
! 238: EXTERNAL DCOPY, XERBLA, ZGGSVP, ZTGSJA
! 239: * ..
! 240: * .. Intrinsic Functions ..
! 241: INTRINSIC MAX, MIN
! 242: * ..
! 243: * .. Executable Statements ..
! 244: *
! 245: * Decode and test the input parameters
! 246: *
! 247: WANTU = LSAME( JOBU, 'U' )
! 248: WANTV = LSAME( JOBV, 'V' )
! 249: WANTQ = LSAME( JOBQ, 'Q' )
! 250: *
! 251: INFO = 0
! 252: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
! 253: INFO = -1
! 254: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
! 255: INFO = -2
! 256: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
! 257: INFO = -3
! 258: ELSE IF( M.LT.0 ) THEN
! 259: INFO = -4
! 260: ELSE IF( N.LT.0 ) THEN
! 261: INFO = -5
! 262: ELSE IF( P.LT.0 ) THEN
! 263: INFO = -6
! 264: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 265: INFO = -10
! 266: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
! 267: INFO = -12
! 268: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
! 269: INFO = -16
! 270: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
! 271: INFO = -18
! 272: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
! 273: INFO = -20
! 274: END IF
! 275: IF( INFO.NE.0 ) THEN
! 276: CALL XERBLA( 'ZGGSVD', -INFO )
! 277: RETURN
! 278: END IF
! 279: *
! 280: * Compute the Frobenius norm of matrices A and B
! 281: *
! 282: ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
! 283: BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
! 284: *
! 285: * Get machine precision and set up threshold for determining
! 286: * the effective numerical rank of the matrices A and B.
! 287: *
! 288: ULP = DLAMCH( 'Precision' )
! 289: UNFL = DLAMCH( 'Safe Minimum' )
! 290: TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
! 291: TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
! 292: *
! 293: CALL ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
! 294: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
! 295: $ WORK, WORK( N+1 ), INFO )
! 296: *
! 297: * Compute the GSVD of two upper "triangular" matrices
! 298: *
! 299: CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
! 300: $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
! 301: $ WORK, NCYCLE, INFO )
! 302: *
! 303: * Sort the singular values and store the pivot indices in IWORK
! 304: * Copy ALPHA to RWORK, then sort ALPHA in RWORK
! 305: *
! 306: CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
! 307: IBND = MIN( L, M-K )
! 308: DO 20 I = 1, IBND
! 309: *
! 310: * Scan for largest ALPHA(K+I)
! 311: *
! 312: ISUB = I
! 313: SMAX = RWORK( K+I )
! 314: DO 10 J = I + 1, IBND
! 315: TEMP = RWORK( K+J )
! 316: IF( TEMP.GT.SMAX ) THEN
! 317: ISUB = J
! 318: SMAX = TEMP
! 319: END IF
! 320: 10 CONTINUE
! 321: IF( ISUB.NE.I ) THEN
! 322: RWORK( K+ISUB ) = RWORK( K+I )
! 323: RWORK( K+I ) = SMAX
! 324: IWORK( K+I ) = K + ISUB
! 325: ELSE
! 326: IWORK( K+I ) = K + I
! 327: END IF
! 328: 20 CONTINUE
! 329: *
! 330: RETURN
! 331: *
! 332: * End of ZGGSVD
! 333: *
! 334: END
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