#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zppsvx_(char *fact, char *uplo, integer *n, integer * nrhs, doublecomplex *ap, doublecomplex *afp, char *equed, doublereal * s, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *rcond, doublereal *ferr, doublereal *berr, doublecomplex * work, doublereal *rwork, integer *info ) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also provided. Description =========== The following steps are performed: 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U'* U , if UPLO = 'U', or A = L * L', if UPLO = 'L', where U is an upper triangular matrix, L is a lower triangular matrix, and ' indicates conjugate transpose. 3. If the leading i-by-i principal minor is not positive definite, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. 4. The system of equations is solved for X using the factored form of A. 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration. Arguments ========= FACT (input) CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified. = 'N': The matrix A will be copied to AFP and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AFP and factored. UPLO (input) CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. See below for further details. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S). AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A. If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the original matrix A. If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix). EQUED (input or output) CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. S (input or output) DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) COMPLEX*16 array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. FERR (output) DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. BERR (output) DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). WORK (workspace) COMPLEX*16 array, dimension (2*N) RWORK (workspace) DOUBLE PRECISION array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. Further Details =============== The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U': Two-dimensional storage of the Hermitian matrix A: a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = conjg(aji)) a44 Packed storage of the upper triangle of A: AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ===================================================================== Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; /* System generated locals */ integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5; doublereal d__1, d__2; doublecomplex z__1; /* Local variables */ static integer i__, j; static doublereal amax, smin, smax; extern logical lsame_(char *, char *); static doublereal scond, anorm; static logical equil, rcequ; extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *); extern doublereal dlamch_(char *); static logical nofact; extern /* Subroutine */ int xerbla_(char *, integer *); static doublereal bignum; static integer infequ; extern doublereal zlanhp_(char *, char *, integer *, doublecomplex *, doublereal *); extern /* Subroutine */ int zlaqhp_(char *, integer *, doublecomplex *, doublereal *, doublereal *, doublereal *, char *), zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zppcon_(char *, integer *, doublecomplex *, doublereal *, doublereal *, doublecomplex *, doublereal *, integer *); static doublereal smlnum; extern /* Subroutine */ int zppequ_(char *, integer *, doublecomplex *, doublereal *, doublereal *, doublereal *, integer *), zpprfs_(char *, integer *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublecomplex *, doublereal *, integer *), zpptrf_(char *, integer *, doublecomplex *, integer *), zpptrs_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, integer *); --ap; --afp; --s; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; x_dim1 = *ldx; x_offset = 1 + x_dim1; x -= x_offset; --ferr; --berr; --work; --rwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); if (nofact || equil) { *(unsigned char *)equed = 'N'; rcequ = FALSE_; } else { rcequ = lsame_(equed, "Y"); smlnum = dlamch_("Safe minimum"); bignum = 1. / smlnum; } /* Test the input parameters. */ if (! nofact && ! equil && ! lsame_(fact, "F")) { *info = -1; } else if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*nrhs < 0) { *info = -4; } else if (lsame_(fact, "F") && ! (rcequ || lsame_( equed, "N"))) { *info = -7; } else { if (rcequ) { smin = bignum; smax = 0.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = smin, d__2 = s[j]; smin = min(d__1,d__2); /* Computing MAX */ d__1 = smax, d__2 = s[j]; smax = max(d__1,d__2); /* L10: */ } if (smin <= 0.) { *info = -8; } else if (*n > 0) { scond = max(smin,smlnum) / min(smax,bignum); } else { scond = 1.; } } if (*info == 0) { if (*ldb < max(1,*n)) { *info = -10; } else if (*ldx < max(1,*n)) { *info = -12; } } } if (*info != 0) { i__1 = -(*info); xerbla_("ZPPSVX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ zppequ_(uplo, n, &ap[1], &s[1], &scond, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ zlaqhp_(uplo, n, &ap[1], &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right-hand side. */ if (rcequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__3 = i__ + j * b_dim1; i__4 = i__; i__5 = i__ + j * b_dim1; z__1.r = s[i__4] * b[i__5].r, z__1.i = s[i__4] * b[i__5].i; b[i__3].r = z__1.r, b[i__3].i = z__1.i; /* L20: */ } /* L30: */ } } if (nofact || equil) { /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ i__1 = *n * (*n + 1) / 2; zcopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1); zpptrf_(uplo, n, &afp[1], info); /* Return if INFO is non-zero. */ if (*info > 0) { *rcond = 0.; return 0; } } /* Compute the norm of the matrix A. */ anorm = zlanhp_("I", uplo, n, &ap[1], &rwork[1]); /* Compute the reciprocal of the condition number of A. */ zppcon_(uplo, n, &afp[1], &anorm, rcond, &work[1], &rwork[1], info); /* Compute the solution matrix X. */ zlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); zpptrs_(uplo, n, nrhs, &afp[1], &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and compute error bounds and backward error estimates for it. */ zpprfs_(uplo, n, nrhs, &ap[1], &afp[1], &b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1], &rwork[1], info); /* Transform the solution matrix X to a solution of the original system. */ if (rcequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__3 = i__ + j * x_dim1; i__4 = i__; i__5 = i__ + j * x_dim1; z__1.r = s[i__4] * x[i__5].r, z__1.i = s[i__4] * x[i__5].i; x[i__3].r = z__1.r, x[i__3].i = z__1.i; /* L40: */ } /* L50: */ } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] /= scond; /* L60: */ } } /* Set INFO = N+1 if the matrix is singular to working precision. */ if (*rcond < dlamch_("Epsilon")) { *info = *n + 1; } return 0; /* End of ZPPSVX */ } /* zppsvx_ */