Annotation of rpl/lapack/lapack/dggglm.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
! 2: $ INFO )
! 3: *
! 4: * -- LAPACK driver routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: INTEGER INFO, LDA, LDB, LWORK, M, N, P
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
! 14: $ X( * ), Y( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DGGGLM solves a general Gauss-Markov linear model (GLM) problem:
! 21: *
! 22: * minimize || y ||_2 subject to d = A*x + B*y
! 23: * x
! 24: *
! 25: * where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
! 26: * given N-vector. It is assumed that M <= N <= M+P, and
! 27: *
! 28: * rank(A) = M and rank( A B ) = N.
! 29: *
! 30: * Under these assumptions, the constrained equation is always
! 31: * consistent, and there is a unique solution x and a minimal 2-norm
! 32: * solution y, which is obtained using a generalized QR factorization
! 33: * of the matrices (A, B) given by
! 34: *
! 35: * A = Q*(R), B = Q*T*Z.
! 36: * (0)
! 37: *
! 38: * In particular, if matrix B is square nonsingular, then the problem
! 39: * GLM is equivalent to the following weighted linear least squares
! 40: * problem
! 41: *
! 42: * minimize || inv(B)*(d-A*x) ||_2
! 43: * x
! 44: *
! 45: * where inv(B) denotes the inverse of B.
! 46: *
! 47: * Arguments
! 48: * =========
! 49: *
! 50: * N (input) INTEGER
! 51: * The number of rows of the matrices A and B. N >= 0.
! 52: *
! 53: * M (input) INTEGER
! 54: * The number of columns of the matrix A. 0 <= M <= N.
! 55: *
! 56: * P (input) INTEGER
! 57: * The number of columns of the matrix B. P >= N-M.
! 58: *
! 59: * A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
! 60: * On entry, the N-by-M matrix A.
! 61: * On exit, the upper triangular part of the array A contains
! 62: * the M-by-M upper triangular matrix R.
! 63: *
! 64: * LDA (input) INTEGER
! 65: * The leading dimension of the array A. LDA >= max(1,N).
! 66: *
! 67: * B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
! 68: * On entry, the N-by-P matrix B.
! 69: * On exit, if N <= P, the upper triangle of the subarray
! 70: * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
! 71: * if N > P, the elements on and above the (N-P)th subdiagonal
! 72: * contain the N-by-P upper trapezoidal matrix T.
! 73: *
! 74: * LDB (input) INTEGER
! 75: * The leading dimension of the array B. LDB >= max(1,N).
! 76: *
! 77: * D (input/output) DOUBLE PRECISION array, dimension (N)
! 78: * On entry, D is the left hand side of the GLM equation.
! 79: * On exit, D is destroyed.
! 80: *
! 81: * X (output) DOUBLE PRECISION array, dimension (M)
! 82: * Y (output) DOUBLE PRECISION array, dimension (P)
! 83: * On exit, X and Y are the solutions of the GLM problem.
! 84: *
! 85: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 86: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 87: *
! 88: * LWORK (input) INTEGER
! 89: * The dimension of the array WORK. LWORK >= max(1,N+M+P).
! 90: * For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
! 91: * where NB is an upper bound for the optimal blocksizes for
! 92: * DGEQRF, SGERQF, DORMQR and SORMRQ.
! 93: *
! 94: * If LWORK = -1, then a workspace query is assumed; the routine
! 95: * only calculates the optimal size of the WORK array, returns
! 96: * this value as the first entry of the WORK array, and no error
! 97: * message related to LWORK is issued by XERBLA.
! 98: *
! 99: * INFO (output) INTEGER
! 100: * = 0: successful exit.
! 101: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 102: * = 1: the upper triangular factor R associated with A in the
! 103: * generalized QR factorization of the pair (A, B) is
! 104: * singular, so that rank(A) < M; the least squares
! 105: * solution could not be computed.
! 106: * = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal
! 107: * factor T associated with B in the generalized QR
! 108: * factorization of the pair (A, B) is singular, so that
! 109: * rank( A B ) < N; the least squares solution could not
! 110: * be computed.
! 111: *
! 112: * ===================================================================
! 113: *
! 114: * .. Parameters ..
! 115: DOUBLE PRECISION ZERO, ONE
! 116: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 117: * ..
! 118: * .. Local Scalars ..
! 119: LOGICAL LQUERY
! 120: INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
! 121: $ NB4, NP
! 122: * ..
! 123: * .. External Subroutines ..
! 124: EXTERNAL DCOPY, DGEMV, DGGQRF, DORMQR, DORMRQ, DTRTRS,
! 125: $ XERBLA
! 126: * ..
! 127: * .. External Functions ..
! 128: INTEGER ILAENV
! 129: EXTERNAL ILAENV
! 130: * ..
! 131: * .. Intrinsic Functions ..
! 132: INTRINSIC INT, MAX, MIN
! 133: * ..
! 134: * .. Executable Statements ..
! 135: *
! 136: * Test the input parameters
! 137: *
! 138: INFO = 0
! 139: NP = MIN( N, P )
! 140: LQUERY = ( LWORK.EQ.-1 )
! 141: IF( N.LT.0 ) THEN
! 142: INFO = -1
! 143: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
! 144: INFO = -2
! 145: ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN
! 146: INFO = -3
! 147: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 148: INFO = -5
! 149: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 150: INFO = -7
! 151: END IF
! 152: *
! 153: * Calculate workspace
! 154: *
! 155: IF( INFO.EQ.0) THEN
! 156: IF( N.EQ.0 ) THEN
! 157: LWKMIN = 1
! 158: LWKOPT = 1
! 159: ELSE
! 160: NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
! 161: NB2 = ILAENV( 1, 'DGERQF', ' ', N, M, -1, -1 )
! 162: NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
! 163: NB4 = ILAENV( 1, 'DORMRQ', ' ', N, M, P, -1 )
! 164: NB = MAX( NB1, NB2, NB3, NB4 )
! 165: LWKMIN = M + N + P
! 166: LWKOPT = M + NP + MAX( N, P )*NB
! 167: END IF
! 168: WORK( 1 ) = LWKOPT
! 169: *
! 170: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
! 171: INFO = -12
! 172: END IF
! 173: END IF
! 174: *
! 175: IF( INFO.NE.0 ) THEN
! 176: CALL XERBLA( 'DGGGLM', -INFO )
! 177: RETURN
! 178: ELSE IF( LQUERY ) THEN
! 179: RETURN
! 180: END IF
! 181: *
! 182: * Quick return if possible
! 183: *
! 184: IF( N.EQ.0 )
! 185: $ RETURN
! 186: *
! 187: * Compute the GQR factorization of matrices A and B:
! 188: *
! 189: * Q'*A = ( R11 ) M, Q'*B*Z' = ( T11 T12 ) M
! 190: * ( 0 ) N-M ( 0 T22 ) N-M
! 191: * M M+P-N N-M
! 192: *
! 193: * where R11 and T22 are upper triangular, and Q and Z are
! 194: * orthogonal.
! 195: *
! 196: CALL DGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
! 197: $ WORK( M+NP+1 ), LWORK-M-NP, INFO )
! 198: LOPT = WORK( M+NP+1 )
! 199: *
! 200: * Update left-hand-side vector d = Q'*d = ( d1 ) M
! 201: * ( d2 ) N-M
! 202: *
! 203: CALL DORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D,
! 204: $ MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
! 205: LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) )
! 206: *
! 207: * Solve T22*y2 = d2 for y2
! 208: *
! 209: IF( N.GT.M ) THEN
! 210: CALL DTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1,
! 211: $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO )
! 212: *
! 213: IF( INFO.GT.0 ) THEN
! 214: INFO = 1
! 215: RETURN
! 216: END IF
! 217: *
! 218: CALL DCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 )
! 219: END IF
! 220: *
! 221: * Set y1 = 0
! 222: *
! 223: DO 10 I = 1, M + P - N
! 224: Y( I ) = ZERO
! 225: 10 CONTINUE
! 226: *
! 227: * Update d1 = d1 - T12*y2
! 228: *
! 229: CALL DGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB,
! 230: $ Y( M+P-N+1 ), 1, ONE, D, 1 )
! 231: *
! 232: * Solve triangular system: R11*x = d1
! 233: *
! 234: IF( M.GT.0 ) THEN
! 235: CALL DTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA,
! 236: $ D, M, INFO )
! 237: *
! 238: IF( INFO.GT.0 ) THEN
! 239: INFO = 2
! 240: RETURN
! 241: END IF
! 242: *
! 243: * Copy D to X
! 244: *
! 245: CALL DCOPY( M, D, 1, X, 1 )
! 246: END IF
! 247: *
! 248: * Backward transformation y = Z'*y
! 249: *
! 250: CALL DORMRQ( 'Left', 'Transpose', P, 1, NP,
! 251: $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
! 252: $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
! 253: WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) )
! 254: *
! 255: RETURN
! 256: *
! 257: * End of DGGGLM
! 258: *
! 259: END
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