#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dlaqtr_(logical *ltran, logical *lreal, integer *n, doublereal *t, integer *ldt, doublereal *b, doublereal *w, doublereal *scale, doublereal *x, doublereal *work, integer *info) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DLAQTR solves the real quasi-triangular system op(T)*p = scale*c, if LREAL = .TRUE. or the complex quasi-triangular systems op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE. in real arithmetic, where T is upper quasi-triangular. If LREAL = .FALSE., then the first diagonal block of T must be 1 by 1, B is the specially structured matrix B = [ b(1) b(2) ... b(n) ] [ w ] [ w ] [ . ] [ w ] op(A) = A or A', A' denotes the conjugate transpose of matrix A. On input, X = [ c ]. On output, X = [ p ]. [ d ] [ q ] This subroutine is designed for the condition number estimation in routine DTRSNA. Arguments ========= LTRAN (input) LOGICAL On entry, LTRAN specifies the option of conjugate transpose: = .FALSE., op(T+i*B) = T+i*B, = .TRUE., op(T+i*B) = (T+i*B)'. LREAL (input) LOGICAL On entry, LREAL specifies the input matrix structure: = .FALSE., the input is complex = .TRUE., the input is real N (input) INTEGER On entry, N specifies the order of T+i*B. N >= 0. T (input) DOUBLE PRECISION array, dimension (LDT,N) On entry, T contains a matrix in Schur canonical form. If LREAL = .FALSE., then the first diagonal block of T mu be 1 by 1. LDT (input) INTEGER The leading dimension of the matrix T. LDT >= max(1,N). B (input) DOUBLE PRECISION array, dimension (N) On entry, B contains the elements to form the matrix B as described above. If LREAL = .TRUE., B is not referenced. W (input) DOUBLE PRECISION On entry, W is the diagonal element of the matrix B. If LREAL = .TRUE., W is not referenced. SCALE (output) DOUBLE PRECISION On exit, SCALE is the scale factor. X (input/output) DOUBLE PRECISION array, dimension (2*N) On entry, X contains the right hand side of the system. On exit, X is overwritten by the solution. WORK (workspace) DOUBLE PRECISION array, dimension (N) INFO (output) INTEGER On exit, INFO is set to 0: successful exit. 1: the some diagonal 1 by 1 block has been perturbed by a small number SMIN to keep nonsingularity. 2: the some diagonal 2 by 2 block has been perturbed by a small number in DLALN2 to keep nonsingularity. NOTE: In the interests of speed, this routine does not check the inputs for errors. ===================================================================== Do not test the input parameters for errors Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static logical c_false = FALSE_; static integer c__2 = 2; static doublereal c_b21 = 1.; static doublereal c_b25 = 0.; static logical c_true = TRUE_; /* System generated locals */ integer t_dim1, t_offset, i__1, i__2; doublereal d__1, d__2, d__3, d__4, d__5, d__6; /* Local variables */ static doublereal d__[4] /* was [2][2] */; static integer i__, j, k; static doublereal v[4] /* was [2][2] */, z__; static integer j1, j2, n1, n2; static doublereal si, xj, sr, rec, eps, tjj, tmp; extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); static integer ierr; static doublereal smin, xmax; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); extern doublereal dasum_(integer *, doublereal *, integer *); extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static integer jnext; static doublereal sminw, xnorm; extern /* Subroutine */ int dlaln2_(logical *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal * , doublereal *, integer *, doublereal *, doublereal *, integer *); extern doublereal dlamch_(char *), dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern integer idamax_(integer *, doublereal *, integer *); static doublereal scaloc; extern /* Subroutine */ int dladiv_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); static doublereal bignum; static logical notran; static doublereal smlnum; t_dim1 = *ldt; t_offset = 1 + t_dim1; t -= t_offset; --b; --x; --work; /* Function Body */ notran = ! (*ltran); *info = 0; /* Quick return if possible */ if (*n == 0) { return 0; } /* Set constants to control overflow */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; bignum = 1. / smlnum; xnorm = dlange_("M", n, n, &t[t_offset], ldt, d__); if (! (*lreal)) { /* Computing MAX */ d__1 = xnorm, d__2 = abs(*w), d__1 = max(d__1,d__2), d__2 = dlange_( "M", n, &c__1, &b[1], n, d__); xnorm = max(d__1,d__2); } /* Computing MAX */ d__1 = smlnum, d__2 = eps * xnorm; smin = max(d__1,d__2); /* Compute 1-norm of each column of strictly upper triangular part of T to control overflow in triangular solver. */ work[1] = 0.; i__1 = *n; for (j = 2; j <= i__1; ++j) { i__2 = j - 1; work[j] = dasum_(&i__2, &t[j * t_dim1 + 1], &c__1); /* L10: */ } if (! (*lreal)) { i__1 = *n; for (i__ = 2; i__ <= i__1; ++i__) { work[i__] += (d__1 = b[i__], abs(d__1)); /* L20: */ } } n2 = *n << 1; n1 = *n; if (! (*lreal)) { n1 = n2; } k = idamax_(&n1, &x[1], &c__1); xmax = (d__1 = x[k], abs(d__1)); *scale = 1.; if (xmax > bignum) { *scale = bignum / xmax; dscal_(&n1, scale, &x[1], &c__1); xmax = bignum; } if (*lreal) { if (notran) { /* Solve T*p = scale*c */ jnext = *n; for (j = *n; j >= 1; --j) { if (j > jnext) { goto L30; } j1 = j; j2 = j; jnext = j - 1; if (j > 1) { if (t[j + (j - 1) * t_dim1] != 0.) { j1 = j - 1; jnext = j - 2; } } if (j1 == j2) { /* Meet 1 by 1 diagonal block Scale to avoid overflow when computing x(j) = b(j)/T(j,j) */ xj = (d__1 = x[j1], abs(d__1)); tjj = (d__1 = t[j1 + j1 * t_dim1], abs(d__1)); tmp = t[j1 + j1 * t_dim1]; if (tjj < smin) { tmp = smin; tjj = smin; *info = 1; } if (xj == 0.) { goto L30; } if (tjj < 1.) { if (xj > bignum * tjj) { rec = 1. / xj; dscal_(n, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } x[j1] /= tmp; xj = (d__1 = x[j1], abs(d__1)); /* Scale x if necessary to avoid overflow when adding a multiple of column j1 of T. */ if (xj > 1.) { rec = 1. / xj; if (work[j1] > (bignum - xmax) * rec) { dscal_(n, &rec, &x[1], &c__1); *scale *= rec; } } if (j1 > 1) { i__1 = j1 - 1; d__1 = -x[j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; k = idamax_(&i__1, &x[1], &c__1); xmax = (d__1 = x[k], abs(d__1)); } } else { /* Meet 2 by 2 diagonal block Call 2 by 2 linear system solve, to take care of possible overflow by scaling factor. */ d__[0] = x[j1]; d__[1] = x[j2]; dlaln2_(&c_false, &c__2, &c__1, &smin, &c_b21, &t[j1 + j1 * t_dim1], ldt, &c_b21, &c_b21, d__, &c__2, & c_b25, &c_b25, v, &c__2, &scaloc, &xnorm, &ierr); if (ierr != 0) { *info = 2; } if (scaloc != 1.) { dscal_(n, &scaloc, &x[1], &c__1); *scale *= scaloc; } x[j1] = v[0]; x[j2] = v[1]; /* Scale V(1,1) (= X(J1)) and/or V(2,1) (=X(J2)) to avoid overflow in updating right-hand side. Computing MAX */ d__1 = abs(v[0]), d__2 = abs(v[1]); xj = max(d__1,d__2); if (xj > 1.) { rec = 1. / xj; /* Computing MAX */ d__1 = work[j1], d__2 = work[j2]; if (max(d__1,d__2) > (bignum - xmax) * rec) { dscal_(n, &rec, &x[1], &c__1); *scale *= rec; } } /* Update right-hand side */ if (j1 > 1) { i__1 = j1 - 1; d__1 = -x[j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; d__1 = -x[j2]; daxpy_(&i__1, &d__1, &t[j2 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; k = idamax_(&i__1, &x[1], &c__1); xmax = (d__1 = x[k], abs(d__1)); } } L30: ; } } else { /* Solve T'*p = scale*c */ jnext = 1; i__1 = *n; for (j = 1; j <= i__1; ++j) { if (j < jnext) { goto L40; } j1 = j; j2 = j; jnext = j + 1; if (j < *n) { if (t[j + 1 + j * t_dim1] != 0.) { j2 = j + 1; jnext = j + 2; } } if (j1 == j2) { /* 1 by 1 diagonal block Scale if necessary to avoid overflow in forming the right-hand side element by inner product. */ xj = (d__1 = x[j1], abs(d__1)); if (xmax > 1.) { rec = 1. / xmax; if (work[j1] > (bignum - xj) * rec) { dscal_(n, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } i__2 = j1 - 1; x[j1] -= ddot_(&i__2, &t[j1 * t_dim1 + 1], &c__1, &x[1], & c__1); xj = (d__1 = x[j1], abs(d__1)); tjj = (d__1 = t[j1 + j1 * t_dim1], abs(d__1)); tmp = t[j1 + j1 * t_dim1]; if (tjj < smin) { tmp = smin; tjj = smin; *info = 1; } if (tjj < 1.) { if (xj > bignum * tjj) { rec = 1. / xj; dscal_(n, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } x[j1] /= tmp; /* Computing MAX */ d__2 = xmax, d__3 = (d__1 = x[j1], abs(d__1)); xmax = max(d__2,d__3); } else { /* 2 by 2 diagonal block Scale if necessary to avoid overflow in forming the right-hand side elements by inner product. Computing MAX */ d__3 = (d__1 = x[j1], abs(d__1)), d__4 = (d__2 = x[j2], abs(d__2)); xj = max(d__3,d__4); if (xmax > 1.) { rec = 1. / xmax; /* Computing MAX */ d__1 = work[j2], d__2 = work[j1]; if (max(d__1,d__2) > (bignum - xj) * rec) { dscal_(n, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } i__2 = j1 - 1; d__[0] = x[j1] - ddot_(&i__2, &t[j1 * t_dim1 + 1], &c__1, &x[1], &c__1); i__2 = j1 - 1; d__[1] = x[j2] - ddot_(&i__2, &t[j2 * t_dim1 + 1], &c__1, &x[1], &c__1); dlaln2_(&c_true, &c__2, &c__1, &smin, &c_b21, &t[j1 + j1 * t_dim1], ldt, &c_b21, &c_b21, d__, &c__2, &c_b25, &c_b25, v, &c__2, &scaloc, &xnorm, &ierr); if (ierr != 0) { *info = 2; } if (scaloc != 1.) { dscal_(n, &scaloc, &x[1], &c__1); *scale *= scaloc; } x[j1] = v[0]; x[j2] = v[1]; /* Computing MAX */ d__3 = (d__1 = x[j1], abs(d__1)), d__4 = (d__2 = x[j2], abs(d__2)), d__3 = max(d__3,d__4); xmax = max(d__3,xmax); } L40: ; } } } else { /* Computing MAX */ d__1 = eps * abs(*w); sminw = max(d__1,smin); if (notran) { /* Solve (T + iB)*(p+iq) = c+id */ jnext = *n; for (j = *n; j >= 1; --j) { if (j > jnext) { goto L70; } j1 = j; j2 = j; jnext = j - 1; if (j > 1) { if (t[j + (j - 1) * t_dim1] != 0.) { j1 = j - 1; jnext = j - 2; } } if (j1 == j2) { /* 1 by 1 diagonal block Scale if necessary to avoid overflow in division */ z__ = *w; if (j1 == 1) { z__ = b[1]; } xj = (d__1 = x[j1], abs(d__1)) + (d__2 = x[*n + j1], abs( d__2)); tjj = (d__1 = t[j1 + j1 * t_dim1], abs(d__1)) + abs(z__); tmp = t[j1 + j1 * t_dim1]; if (tjj < sminw) { tmp = sminw; tjj = sminw; *info = 1; } if (xj == 0.) { goto L70; } if (tjj < 1.) { if (xj > bignum * tjj) { rec = 1. / xj; dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } dladiv_(&x[j1], &x[*n + j1], &tmp, &z__, &sr, &si); x[j1] = sr; x[*n + j1] = si; xj = (d__1 = x[j1], abs(d__1)) + (d__2 = x[*n + j1], abs( d__2)); /* Scale x if necessary to avoid overflow when adding a multiple of column j1 of T. */ if (xj > 1.) { rec = 1. / xj; if (work[j1] > (bignum - xmax) * rec) { dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; } } if (j1 > 1) { i__1 = j1 - 1; d__1 = -x[j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; d__1 = -x[*n + j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[* n + 1], &c__1); x[1] += b[j1] * x[*n + j1]; x[*n + 1] -= b[j1] * x[j1]; xmax = 0.; i__1 = j1 - 1; for (k = 1; k <= i__1; ++k) { /* Computing MAX */ d__3 = xmax, d__4 = (d__1 = x[k], abs(d__1)) + ( d__2 = x[k + *n], abs(d__2)); xmax = max(d__3,d__4); /* L50: */ } } } else { /* Meet 2 by 2 diagonal block */ d__[0] = x[j1]; d__[1] = x[j2]; d__[2] = x[*n + j1]; d__[3] = x[*n + j2]; d__1 = -(*w); dlaln2_(&c_false, &c__2, &c__2, &sminw, &c_b21, &t[j1 + j1 * t_dim1], ldt, &c_b21, &c_b21, d__, &c__2, & c_b25, &d__1, v, &c__2, &scaloc, &xnorm, &ierr); if (ierr != 0) { *info = 2; } if (scaloc != 1.) { i__1 = *n << 1; dscal_(&i__1, &scaloc, &x[1], &c__1); *scale = scaloc * *scale; } x[j1] = v[0]; x[j2] = v[1]; x[*n + j1] = v[2]; x[*n + j2] = v[3]; /* Scale X(J1), .... to avoid overflow in updating right hand side. Computing MAX */ d__1 = abs(v[0]) + abs(v[2]), d__2 = abs(v[1]) + abs(v[3]) ; xj = max(d__1,d__2); if (xj > 1.) { rec = 1. / xj; /* Computing MAX */ d__1 = work[j1], d__2 = work[j2]; if (max(d__1,d__2) > (bignum - xmax) * rec) { dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; } } /* Update the right-hand side. */ if (j1 > 1) { i__1 = j1 - 1; d__1 = -x[j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; d__1 = -x[j2]; daxpy_(&i__1, &d__1, &t[j2 * t_dim1 + 1], &c__1, &x[1] , &c__1); i__1 = j1 - 1; d__1 = -x[*n + j1]; daxpy_(&i__1, &d__1, &t[j1 * t_dim1 + 1], &c__1, &x[* n + 1], &c__1); i__1 = j1 - 1; d__1 = -x[*n + j2]; daxpy_(&i__1, &d__1, &t[j2 * t_dim1 + 1], &c__1, &x[* n + 1], &c__1); x[1] = x[1] + b[j1] * x[*n + j1] + b[j2] * x[*n + j2]; x[*n + 1] = x[*n + 1] - b[j1] * x[j1] - b[j2] * x[j2]; xmax = 0.; i__1 = j1 - 1; for (k = 1; k <= i__1; ++k) { /* Computing MAX */ d__3 = (d__1 = x[k], abs(d__1)) + (d__2 = x[k + * n], abs(d__2)); xmax = max(d__3,xmax); /* L60: */ } } } L70: ; } } else { /* Solve (T + iB)'*(p+iq) = c+id */ jnext = 1; i__1 = *n; for (j = 1; j <= i__1; ++j) { if (j < jnext) { goto L80; } j1 = j; j2 = j; jnext = j + 1; if (j < *n) { if (t[j + 1 + j * t_dim1] != 0.) { j2 = j + 1; jnext = j + 2; } } if (j1 == j2) { /* 1 by 1 diagonal block Scale if necessary to avoid overflow in forming the right-hand side element by inner product. */ xj = (d__1 = x[j1], abs(d__1)) + (d__2 = x[j1 + *n], abs( d__2)); if (xmax > 1.) { rec = 1. / xmax; if (work[j1] > (bignum - xj) * rec) { dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } i__2 = j1 - 1; x[j1] -= ddot_(&i__2, &t[j1 * t_dim1 + 1], &c__1, &x[1], & c__1); i__2 = j1 - 1; x[*n + j1] -= ddot_(&i__2, &t[j1 * t_dim1 + 1], &c__1, &x[ *n + 1], &c__1); if (j1 > 1) { x[j1] -= b[j1] * x[*n + 1]; x[*n + j1] += b[j1] * x[1]; } xj = (d__1 = x[j1], abs(d__1)) + (d__2 = x[j1 + *n], abs( d__2)); z__ = *w; if (j1 == 1) { z__ = b[1]; } /* Scale if necessary to avoid overflow in complex division */ tjj = (d__1 = t[j1 + j1 * t_dim1], abs(d__1)) + abs(z__); tmp = t[j1 + j1 * t_dim1]; if (tjj < sminw) { tmp = sminw; tjj = sminw; *info = 1; } if (tjj < 1.) { if (xj > bignum * tjj) { rec = 1. / xj; dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } d__1 = -z__; dladiv_(&x[j1], &x[*n + j1], &tmp, &d__1, &sr, &si); x[j1] = sr; x[j1 + *n] = si; /* Computing MAX */ d__3 = (d__1 = x[j1], abs(d__1)) + (d__2 = x[j1 + *n], abs(d__2)); xmax = max(d__3,xmax); } else { /* 2 by 2 diagonal block Scale if necessary to avoid overflow in forming the right-hand side element by inner product. Computing MAX */ d__5 = (d__1 = x[j1], abs(d__1)) + (d__2 = x[*n + j1], abs(d__2)), d__6 = (d__3 = x[j2], abs(d__3)) + ( d__4 = x[*n + j2], abs(d__4)); xj = max(d__5,d__6); if (xmax > 1.) { rec = 1. / xmax; /* Computing MAX */ d__1 = work[j1], d__2 = work[j2]; if (max(d__1,d__2) > (bignum - xj) / xmax) { dscal_(&n2, &rec, &x[1], &c__1); *scale *= rec; xmax *= rec; } } i__2 = j1 - 1; d__[0] = x[j1] - ddot_(&i__2, &t[j1 * t_dim1 + 1], &c__1, &x[1], &c__1); i__2 = j1 - 1; d__[1] = x[j2] - ddot_(&i__2, &t[j2 * t_dim1 + 1], &c__1, &x[1], &c__1); i__2 = j1 - 1; d__[2] = x[*n + j1] - ddot_(&i__2, &t[j1 * t_dim1 + 1], & c__1, &x[*n + 1], &c__1); i__2 = j1 - 1; d__[3] = x[*n + j2] - ddot_(&i__2, &t[j2 * t_dim1 + 1], & c__1, &x[*n + 1], &c__1); d__[0] -= b[j1] * x[*n + 1]; d__[1] -= b[j2] * x[*n + 1]; d__[2] += b[j1] * x[1]; d__[3] += b[j2] * x[1]; dlaln2_(&c_true, &c__2, &c__2, &sminw, &c_b21, &t[j1 + j1 * t_dim1], ldt, &c_b21, &c_b21, d__, &c__2, & c_b25, w, v, &c__2, &scaloc, &xnorm, &ierr); if (ierr != 0) { *info = 2; } if (scaloc != 1.) { dscal_(&n2, &scaloc, &x[1], &c__1); *scale = scaloc * *scale; } x[j1] = v[0]; x[j2] = v[1]; x[*n + j1] = v[2]; x[*n + j2] = v[3]; /* Computing MAX */ d__5 = (d__1 = x[j1], abs(d__1)) + (d__2 = x[*n + j1], abs(d__2)), d__6 = (d__3 = x[j2], abs(d__3)) + ( d__4 = x[*n + j2], abs(d__4)), d__5 = max(d__5, d__6); xmax = max(d__5,xmax); } L80: ; } } } return 0; /* End of DLAQTR */ } /* dlaqtr_ */