#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zggglm_(integer *n, integer *m, integer *p, doublecomplex *a, integer *lda, doublecomplex *b, integer *ldb, doublecomplex *d__, doublecomplex *x, doublecomplex *y, doublecomplex *work, integer *lwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZGGGLM solves a general Gauss-Markov linear model (GLM) problem: minimize || y ||_2 subject to d = A*x + B*y x where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. It is assumed that M <= N <= M+P, and rank(A) = M and rank( A B ) = N. Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B) given by A = Q*(R), B = Q*T*Z. (0) In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem minimize || inv(B)*(d-A*x) ||_2 x where inv(B) denotes the inverse of B. Arguments ========= N (input) INTEGER The number of rows of the matrices A and B. N >= 0. M (input) INTEGER The number of columns of the matrix A. 0 <= M <= N. P (input) INTEGER The number of columns of the matrix B. P >= N-M. A (input/output) COMPLEX*16 array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). B (input/output) COMPLEX*16 array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). D (input/output) COMPLEX*16 array, dimension (N) On entry, D is the left hand side of the GLM equation. On exit, D is destroyed. X (output) COMPLEX*16 array, dimension (M) Y (output) COMPLEX*16 array, dimension (P) On exit, X and Y are the solutions of the GLM problem. WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,N+M+P). For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB is an upper bound for the optimal blocksizes for ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ. 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. = 1: the upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed. =================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static doublecomplex c_b2 = {1.,0.}; static integer c__1 = 1; static integer c_n1 = -1; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4; doublecomplex z__1; /* Local variables */ static integer i__, nb, np, nb1, nb2, nb3, nb4, lopt; extern /* Subroutine */ int zgemv_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int zggqrf_(integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer *) ; static integer lwkmin, lwkopt; static logical lquery; extern /* Subroutine */ int zunmqr_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *), zunmrq_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *), ztrtrs_(char *, char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --d__; --x; --y; --work; /* Function Body */ *info = 0; np = min(*n,*p); lquery = *lwork == -1; if (*n < 0) { *info = -1; } else if (*m < 0 || *m > *n) { *info = -2; } else if (*p < 0 || *p < *n - *m) { *info = -3; } else if (*lda < max(1,*n)) { *info = -5; } else if (*ldb < max(1,*n)) { *info = -7; } /* Calculate workspace */ if (*info == 0) { if (*n == 0) { lwkmin = 1; lwkopt = 1; } else { nb1 = ilaenv_(&c__1, "ZGEQRF", " ", n, m, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); nb2 = ilaenv_(&c__1, "ZGERQF", " ", n, m, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); nb3 = ilaenv_(&c__1, "ZUNMQR", " ", n, m, p, &c_n1, (ftnlen)6, ( ftnlen)1); nb4 = ilaenv_(&c__1, "ZUNMRQ", " ", n, m, p, &c_n1, (ftnlen)6, ( ftnlen)1); /* Computing MAX */ i__1 = max(nb1,nb2), i__1 = max(i__1,nb3); nb = max(i__1,nb4); lwkmin = *m + *n + *p; lwkopt = *m + np + max(*n,*p) * nb; } work[1].r = (doublereal) lwkopt, work[1].i = 0.; if (*lwork < lwkmin && ! lquery) { *info = -12; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZGGGLM", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Compute the GQR factorization of matrices A and B: Q'*A = ( R11 ) M, Q'*B*Z' = ( T11 T12 ) M ( 0 ) N-M ( 0 T22 ) N-M M M+P-N N-M where R11 and T22 are upper triangular, and Q and Z are unitary. */ i__1 = *lwork - *m - np; zggqrf_(n, m, p, &a[a_offset], lda, &work[1], &b[b_offset], ldb, &work[*m + 1], &work[*m + np + 1], &i__1, info); i__1 = *m + np + 1; lopt = (integer) work[i__1].r; /* Update left-hand-side vector d = Q'*d = ( d1 ) M ( d2 ) N-M */ i__1 = max(1,*n); i__2 = *lwork - *m - np; zunmqr_("Left", "Conjugate transpose", n, &c__1, m, &a[a_offset], lda, & work[1], &d__[1], &i__1, &work[*m + np + 1], &i__2, info); /* Computing MAX */ i__3 = *m + np + 1; i__1 = lopt, i__2 = (integer) work[i__3].r; lopt = max(i__1,i__2); /* Solve T22*y2 = d2 for y2 */ if (*n > *m) { i__1 = *n - *m; i__2 = *n - *m; ztrtrs_("Upper", "No transpose", "Non unit", &i__1, &c__1, &b[*m + 1 + (*m + *p - *n + 1) * b_dim1], ldb, &d__[*m + 1], &i__2, info); if (*info > 0) { *info = 1; return 0; } i__1 = *n - *m; zcopy_(&i__1, &d__[*m + 1], &c__1, &y[*m + *p - *n + 1], &c__1); } /* Set y1 = 0 */ i__1 = *m + *p - *n; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = i__; y[i__2].r = 0., y[i__2].i = 0.; /* L10: */ } /* Update d1 = d1 - T12*y2 */ i__1 = *n - *m; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", m, &i__1, &z__1, &b[(*m + *p - *n + 1) * b_dim1 + 1], ldb, &y[*m + *p - *n + 1], &c__1, &c_b2, &d__[1], &c__1); /* Solve triangular system: R11*x = d1 */ if (*m > 0) { ztrtrs_("Upper", "No Transpose", "Non unit", m, &c__1, &a[a_offset], lda, &d__[1], m, info); if (*info > 0) { *info = 2; return 0; } /* Copy D to X */ zcopy_(m, &d__[1], &c__1, &x[1], &c__1); } /* Backward transformation y = Z'*y Computing MAX */ i__1 = 1, i__2 = *n - *p + 1; i__3 = max(1,*p); i__4 = *lwork - *m - np; zunmrq_("Left", "Conjugate transpose", p, &c__1, &np, &b[max(i__1,i__2) + b_dim1], ldb, &work[*m + 1], &y[1], &i__3, &work[*m + np + 1], & i__4, info); /* Computing MAX */ i__4 = *m + np + 1; i__2 = lopt, i__3 = (integer) work[i__4].r; i__1 = *m + np + max(i__2,i__3); work[1].r = (doublereal) i__1, work[1].i = 0.; return 0; /* End of ZGGGLM */ } /* zggglm_ */