Annotation of rpl/lapack/lapack/zggqrf.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
! 2: $ LWORK, INFO )
! 3: *
! 4: * -- LAPACK 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: COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
! 14: $ WORK( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZGGQRF computes a generalized QR factorization of an N-by-M matrix A
! 21: * and an N-by-P matrix B:
! 22: *
! 23: * A = Q*R, B = Q*T*Z,
! 24: *
! 25: * where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,
! 26: * and R and T assume one of the forms:
! 27: *
! 28: * if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
! 29: * ( 0 ) N-M N M-N
! 30: * M
! 31: *
! 32: * where R11 is upper triangular, and
! 33: *
! 34: * if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
! 35: * P-N N ( T21 ) P
! 36: * P
! 37: *
! 38: * where T12 or T21 is upper triangular.
! 39: *
! 40: * In particular, if B is square and nonsingular, the GQR factorization
! 41: * of A and B implicitly gives the QR factorization of inv(B)*A:
! 42: *
! 43: * inv(B)*A = Z'*(inv(T)*R)
! 44: *
! 45: * where inv(B) denotes the inverse of the matrix B, and Z' denotes the
! 46: * conjugate transpose of matrix Z.
! 47: *
! 48: * Arguments
! 49: * =========
! 50: *
! 51: * N (input) INTEGER
! 52: * The number of rows of the matrices A and B. N >= 0.
! 53: *
! 54: * M (input) INTEGER
! 55: * The number of columns of the matrix A. M >= 0.
! 56: *
! 57: * P (input) INTEGER
! 58: * The number of columns of the matrix B. P >= 0.
! 59: *
! 60: * A (input/output) COMPLEX*16 array, dimension (LDA,M)
! 61: * On entry, the N-by-M matrix A.
! 62: * On exit, the elements on and above the diagonal of the array
! 63: * contain the min(N,M)-by-M upper trapezoidal matrix R (R is
! 64: * upper triangular if N >= M); the elements below the diagonal,
! 65: * with the array TAUA, represent the unitary matrix Q as a
! 66: * product of min(N,M) elementary reflectors (see Further
! 67: * Details).
! 68: *
! 69: * LDA (input) INTEGER
! 70: * The leading dimension of the array A. LDA >= max(1,N).
! 71: *
! 72: * TAUA (output) COMPLEX*16 array, dimension (min(N,M))
! 73: * The scalar factors of the elementary reflectors which
! 74: * represent the unitary matrix Q (see Further Details).
! 75: *
! 76: * B (input/output) COMPLEX*16 array, dimension (LDB,P)
! 77: * On entry, the N-by-P matrix B.
! 78: * On exit, if N <= P, the upper triangle of the subarray
! 79: * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
! 80: * if N > P, the elements on and above the (N-P)-th subdiagonal
! 81: * contain the N-by-P upper trapezoidal matrix T; the remaining
! 82: * elements, with the array TAUB, represent the unitary
! 83: * matrix Z as a product of elementary reflectors (see Further
! 84: * Details).
! 85: *
! 86: * LDB (input) INTEGER
! 87: * The leading dimension of the array B. LDB >= max(1,N).
! 88: *
! 89: * TAUB (output) COMPLEX*16 array, dimension (min(N,P))
! 90: * The scalar factors of the elementary reflectors which
! 91: * represent the unitary matrix Z (see Further Details).
! 92: *
! 93: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
! 94: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 95: *
! 96: * LWORK (input) INTEGER
! 97: * The dimension of the array WORK. LWORK >= max(1,N,M,P).
! 98: * For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
! 99: * where NB1 is the optimal blocksize for the QR factorization
! 100: * of an N-by-M matrix, NB2 is the optimal blocksize for the
! 101: * RQ factorization of an N-by-P matrix, and NB3 is the optimal
! 102: * blocksize for a call of ZUNMQR.
! 103: *
! 104: * If LWORK = -1, then a workspace query is assumed; the routine
! 105: * only calculates the optimal size of the WORK array, returns
! 106: * this value as the first entry of the WORK array, and no error
! 107: * message related to LWORK is issued by XERBLA.
! 108: *
! 109: * INFO (output) INTEGER
! 110: * = 0: successful exit
! 111: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 112: *
! 113: * Further Details
! 114: * ===============
! 115: *
! 116: * The matrix Q is represented as a product of elementary reflectors
! 117: *
! 118: * Q = H(1) H(2) . . . H(k), where k = min(n,m).
! 119: *
! 120: * Each H(i) has the form
! 121: *
! 122: * H(i) = I - taua * v * v'
! 123: *
! 124: * where taua is a complex scalar, and v is a complex vector with
! 125: * v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
! 126: * and taua in TAUA(i).
! 127: * To form Q explicitly, use LAPACK subroutine ZUNGQR.
! 128: * To use Q to update another matrix, use LAPACK subroutine ZUNMQR.
! 129: *
! 130: * The matrix Z is represented as a product of elementary reflectors
! 131: *
! 132: * Z = H(1) H(2) . . . H(k), where k = min(n,p).
! 133: *
! 134: * Each H(i) has the form
! 135: *
! 136: * H(i) = I - taub * v * v'
! 137: *
! 138: * where taub is a complex scalar, and v is a complex vector with
! 139: * v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
! 140: * B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
! 141: * To form Z explicitly, use LAPACK subroutine ZUNGRQ.
! 142: * To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.
! 143: *
! 144: * =====================================================================
! 145: *
! 146: * .. Local Scalars ..
! 147: LOGICAL LQUERY
! 148: INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
! 149: * ..
! 150: * .. External Subroutines ..
! 151: EXTERNAL XERBLA, ZGEQRF, ZGERQF, ZUNMQR
! 152: * ..
! 153: * .. External Functions ..
! 154: INTEGER ILAENV
! 155: EXTERNAL ILAENV
! 156: * ..
! 157: * .. Intrinsic Functions ..
! 158: INTRINSIC INT, MAX, MIN
! 159: * ..
! 160: * .. Executable Statements ..
! 161: *
! 162: * Test the input parameters
! 163: *
! 164: INFO = 0
! 165: NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 )
! 166: NB2 = ILAENV( 1, 'ZGERQF', ' ', N, P, -1, -1 )
! 167: NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 )
! 168: NB = MAX( NB1, NB2, NB3 )
! 169: LWKOPT = MAX( N, M, P )*NB
! 170: WORK( 1 ) = LWKOPT
! 171: LQUERY = ( LWORK.EQ.-1 )
! 172: IF( N.LT.0 ) THEN
! 173: INFO = -1
! 174: ELSE IF( M.LT.0 ) THEN
! 175: INFO = -2
! 176: ELSE IF( P.LT.0 ) THEN
! 177: INFO = -3
! 178: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 179: INFO = -5
! 180: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 181: INFO = -8
! 182: ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
! 183: INFO = -11
! 184: END IF
! 185: IF( INFO.NE.0 ) THEN
! 186: CALL XERBLA( 'ZGGQRF', -INFO )
! 187: RETURN
! 188: ELSE IF( LQUERY ) THEN
! 189: RETURN
! 190: END IF
! 191: *
! 192: * QR factorization of N-by-M matrix A: A = Q*R
! 193: *
! 194: CALL ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
! 195: LOPT = WORK( 1 )
! 196: *
! 197: * Update B := Q'*B.
! 198: *
! 199: CALL ZUNMQR( 'Left', 'Conjugate Transpose', N, P, MIN( N, M ), A,
! 200: $ LDA, TAUA, B, LDB, WORK, LWORK, INFO )
! 201: LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
! 202: *
! 203: * RQ factorization of N-by-P matrix B: B = T*Z.
! 204: *
! 205: CALL ZGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
! 206: WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
! 207: *
! 208: RETURN
! 209: *
! 210: * End of ZGGQRF
! 211: *
! 212: END
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