#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dlatdf_(integer *ijob, integer *n, doublereal *z__, integer *ldz, doublereal *rhs, doublereal *rdsum, doublereal *rdscal, integer *ipiv, integer *jpiv) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DLATDF uses the LU factorization of the n-by-n matrix Z computed by DGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s. b such that the norm of x is as large as possible. On entry RHS = b holds the contribution from earlier solved sub-systems, and on return RHS = x. The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular. Arguments ========= IJOB (input) INTEGER IJOB = 2: First compute an approximative null-vector e of Z using DGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is choosen as either +1 or -1 (Default). N (input) INTEGER The number of columns of the matrix Z. Z (input) DOUBLE PRECISION array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by DGETC2: Z = P * L * U * Q LDZ (input) INTEGER The leading dimension of the array Z. LDA >= max(1, N). RHS (input/output) DOUBLE PRECISION array, dimension N. On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries acoording to the value of IJOB (see above). RDSUM (input/output) DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate under computation by DTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. RDSCAL (input/output) DOUBLE PRECISION On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when DTGSY2 is called by DTGSYL. IPIV (input) INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i). JPIV (input) INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j). Further Details =============== Based on contributions by Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization. [1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. [2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report IMINF-95.05, Departement of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. ===================================================================== Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c_n1 = -1; static doublereal c_b23 = 1.; static doublereal c_b37 = -1.; /* System generated locals */ integer z_dim1, z_offset, i__1, i__2; doublereal d__1; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, k; static doublereal bm, bp, xm[8], xp[8]; extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); static integer info; static doublereal temp, work[32]; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); extern doublereal dasum_(integer *, doublereal *, integer *); static doublereal pmone; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static doublereal sminu; static integer iwork[8]; static doublereal splus; extern /* Subroutine */ int dgesc2_(integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *), dgecon_(char *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *), dlassq_(integer *, doublereal *, integer *, doublereal *, doublereal *), dlaswp_( integer *, doublereal *, integer *, integer *, integer *, integer *, integer *); z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; --rhs; --ipiv; --jpiv; /* Function Body */ if (*ijob != 2) { /* Apply permutations IPIV to RHS */ i__1 = *n - 1; dlaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &ipiv[1], &c__1); /* Solve for L-part choosing RHS either to +1 or -1. */ pmone = -1.; i__1 = *n - 1; for (j = 1; j <= i__1; ++j) { bp = rhs[j] + 1.; bm = rhs[j] - 1.; splus = 1.; /* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and SMIN computed more efficiently than in BSOLVE [1]. */ i__2 = *n - j; splus += ddot_(&i__2, &z__[j + 1 + j * z_dim1], &c__1, &z__[j + 1 + j * z_dim1], &c__1); i__2 = *n - j; sminu = ddot_(&i__2, &z__[j + 1 + j * z_dim1], &c__1, &rhs[j + 1], &c__1); splus *= rhs[j]; if (splus > sminu) { rhs[j] = bp; } else if (sminu > splus) { rhs[j] = bm; } else { /* In this case the updating sums are equal and we can choose RHS(J) +1 or -1. The first time this happens we choose -1, thereafter +1. This is a simple way to get good estimates of matrices like Byers well-known example (see [1]). (Not done in BSOLVE.) */ rhs[j] += pmone; pmone = 1.; } /* Compute the remaining r.h.s. */ temp = -rhs[j]; i__2 = *n - j; daxpy_(&i__2, &temp, &z__[j + 1 + j * z_dim1], &c__1, &rhs[j + 1], &c__1); /* L10: */ } /* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done in BSOLVE and will hopefully give us a better estimate because any ill-conditioning of the original matrix is transfered to U and not to L. U(N, N) is an approximation to sigma_min(LU). */ i__1 = *n - 1; dcopy_(&i__1, &rhs[1], &c__1, xp, &c__1); xp[*n - 1] = rhs[*n] + 1.; rhs[*n] += -1.; splus = 0.; sminu = 0.; for (i__ = *n; i__ >= 1; --i__) { temp = 1. / z__[i__ + i__ * z_dim1]; xp[i__ - 1] *= temp; rhs[i__] *= temp; i__1 = *n; for (k = i__ + 1; k <= i__1; ++k) { xp[i__ - 1] -= xp[k - 1] * (z__[i__ + k * z_dim1] * temp); rhs[i__] -= rhs[k] * (z__[i__ + k * z_dim1] * temp); /* L20: */ } splus += (d__1 = xp[i__ - 1], abs(d__1)); sminu += (d__1 = rhs[i__], abs(d__1)); /* L30: */ } if (splus > sminu) { dcopy_(n, xp, &c__1, &rhs[1], &c__1); } /* Apply the permutations JPIV to the computed solution (RHS) */ i__1 = *n - 1; dlaswp_(&c__1, &rhs[1], ldz, &c__1, &i__1, &jpiv[1], &c_n1); /* Compute the sum of squares */ dlassq_(n, &rhs[1], &c__1, rdscal, rdsum); } else { /* IJOB = 2, Compute approximate nullvector XM of Z */ dgecon_("I", n, &z__[z_offset], ldz, &c_b23, &temp, work, iwork, & info); dcopy_(n, &work[*n], &c__1, xm, &c__1); /* Compute RHS */ i__1 = *n - 1; dlaswp_(&c__1, xm, ldz, &c__1, &i__1, &ipiv[1], &c_n1); temp = 1. / sqrt(ddot_(n, xm, &c__1, xm, &c__1)); dscal_(n, &temp, xm, &c__1); dcopy_(n, xm, &c__1, xp, &c__1); daxpy_(n, &c_b23, &rhs[1], &c__1, xp, &c__1); daxpy_(n, &c_b37, xm, &c__1, &rhs[1], &c__1); dgesc2_(n, &z__[z_offset], ldz, &rhs[1], &ipiv[1], &jpiv[1], &temp); dgesc2_(n, &z__[z_offset], ldz, xp, &ipiv[1], &jpiv[1], &temp); if (dasum_(n, xp, &c__1) > dasum_(n, &rhs[1], &c__1)) { dcopy_(n, xp, &c__1, &rhs[1], &c__1); } /* Compute the sum of squares */ dlassq_(n, &rhs[1], &c__1, rdscal, rdsum); } return 0; /* End of DLATDF */ } /* dlatdf_ */