/* zptrfs.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static doublecomplex c_b16 = {1.,0.}; /* Subroutine */ int zptrfs_(char *uplo, integer *n, integer *nrhs, doublereal *d__, doublecomplex *e, doublereal *df, doublecomplex *ef, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *ferr, doublereal *berr, doublecomplex *work, doublereal * rwork, integer *info) { /* System generated locals */ integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5, i__6; doublereal d__1, d__2, d__3, d__4, d__5, d__6, d__7, d__8, d__9, d__10, d__11, d__12; doublecomplex z__1, z__2, z__3; /* Builtin functions */ double d_imag(doublecomplex *); void d_cnjg(doublecomplex *, doublecomplex *); double z_abs(doublecomplex *); /* Local variables */ integer i__, j; doublereal s; doublecomplex bi, cx, dx, ex; integer ix, nz; doublereal eps, safe1, safe2; extern logical lsame_(char *, char *); integer count; logical upper; extern /* Subroutine */ int zaxpy_(integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *); extern doublereal dlamch_(char *); extern integer idamax_(integer *, doublereal *, integer *); doublereal safmin; extern /* Subroutine */ int xerbla_(char *, integer *); doublereal lstres; extern /* Subroutine */ int zpttrs_(char *, integer *, integer *, doublereal *, doublecomplex *, doublecomplex *, integer *, integer *); /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* ZPTRFS improves the computed solution to a system of linear */ /* equations when the coefficient matrix is Hermitian positive definite */ /* and tridiagonal, and provides error bounds and backward error */ /* estimates for the solution. */ /* Arguments */ /* ========= */ /* UPLO (input) CHARACTER*1 */ /* Specifies whether the superdiagonal or the subdiagonal of the */ /* tridiagonal matrix A is stored and the form of the */ /* factorization: */ /* = 'U': E is the superdiagonal of A, and A = U**H*D*U; */ /* = 'L': E is the subdiagonal of A, and A = L*D*L**H. */ /* (The two forms are equivalent if A is real.) */ /* N (input) INTEGER */ /* 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 matrix B. NRHS >= 0. */ /* D (input) DOUBLE PRECISION array, dimension (N) */ /* The n real diagonal elements of the tridiagonal matrix A. */ /* E (input) COMPLEX*16 array, dimension (N-1) */ /* The (n-1) off-diagonal elements of the tridiagonal matrix A */ /* (see UPLO). */ /* DF (input) DOUBLE PRECISION array, dimension (N) */ /* The n diagonal elements of the diagonal matrix D from */ /* the factorization computed by ZPTTRF. */ /* EF (input) COMPLEX*16 array, dimension (N-1) */ /* The (n-1) off-diagonal elements of the unit bidiagonal */ /* factor U or L from the factorization computed by ZPTTRF */ /* (see UPLO). */ /* B (input) COMPLEX*16 array, dimension (LDB,NRHS) */ /* The right hand side matrix B. */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */ /* On entry, the solution matrix X, as computed by ZPTTRS. */ /* On exit, the improved solution matrix X. */ /* LDX (input) INTEGER */ /* The leading dimension of the array X. LDX >= max(1,N). */ /* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */ /* The 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). */ /* 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 (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 */ /* Internal Parameters */ /* =================== */ /* ITMAX is the maximum number of steps of iterative refinement. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Statement Functions .. */ /* .. */ /* .. Statement Function definitions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; --df; --ef; 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; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*nrhs < 0) { *info = -3; } else if (*ldb < max(1,*n)) { *info = -9; } else if (*ldx < max(1,*n)) { *info = -11; } if (*info != 0) { i__1 = -(*info); xerbla_("ZPTRFS", &i__1); return 0; } /* Quick return if possible */ if (*n == 0 || *nrhs == 0) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] = 0.; berr[j] = 0.; /* L10: */ } return 0; } /* NZ = maximum number of nonzero elements in each row of A, plus 1 */ nz = 4; eps = dlamch_("Epsilon"); safmin = dlamch_("Safe minimum"); safe1 = nz * safmin; safe2 = safe1 / eps; /* Do for each right hand side */ i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { count = 1; lstres = 3.; L20: /* Loop until stopping criterion is satisfied. */ /* Compute residual R = B - A * X. Also compute */ /* abs(A)*abs(x) + abs(b) for use in the backward error bound. */ if (upper) { if (*n == 1) { i__2 = j * b_dim1 + 1; bi.r = b[i__2].r, bi.i = b[i__2].i; i__2 = j * x_dim1 + 1; z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i; dx.r = z__1.r, dx.i = z__1.i; z__1.r = bi.r - dx.r, z__1.i = bi.i - dx.i; work[1].r = z__1.r, work[1].i = z__1.i; rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 = d_imag(&dx), abs(d__4))); } else { i__2 = j * b_dim1 + 1; bi.r = b[i__2].r, bi.i = b[i__2].i; i__2 = j * x_dim1 + 1; z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i; dx.r = z__1.r, dx.i = z__1.i; i__2 = j * x_dim1 + 2; z__1.r = e[1].r * x[i__2].r - e[1].i * x[i__2].i, z__1.i = e[ 1].r * x[i__2].i + e[1].i * x[i__2].r; ex.r = z__1.r, ex.i = z__1.i; z__2.r = bi.r - dx.r, z__2.i = bi.i - dx.i; z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i; work[1].r = z__1.r, work[1].i = z__1.i; i__2 = j * x_dim1 + 2; rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 = d_imag(&dx), abs(d__4))) + ((d__5 = e[1].r, abs(d__5)) + (d__6 = d_imag(&e[1]), abs(d__6))) * ((d__7 = x[ i__2].r, abs(d__7)) + (d__8 = d_imag(&x[j * x_dim1 + 2]), abs(d__8))); i__2 = *n - 1; for (i__ = 2; i__ <= i__2; ++i__) { i__3 = i__ + j * b_dim1; bi.r = b[i__3].r, bi.i = b[i__3].i; d_cnjg(&z__2, &e[i__ - 1]); i__3 = i__ - 1 + j * x_dim1; z__1.r = z__2.r * x[i__3].r - z__2.i * x[i__3].i, z__1.i = z__2.r * x[i__3].i + z__2.i * x[i__3].r; cx.r = z__1.r, cx.i = z__1.i; i__3 = i__; i__4 = i__ + j * x_dim1; z__1.r = d__[i__3] * x[i__4].r, z__1.i = d__[i__3] * x[ i__4].i; dx.r = z__1.r, dx.i = z__1.i; i__3 = i__; i__4 = i__ + 1 + j * x_dim1; z__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i, z__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[ i__4].r; ex.r = z__1.r, ex.i = z__1.i; i__3 = i__; z__3.r = bi.r - cx.r, z__3.i = bi.i - cx.i; z__2.r = z__3.r - dx.r, z__2.i = z__3.i - dx.i; z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i; work[i__3].r = z__1.r, work[i__3].i = z__1.i; i__3 = i__ - 1; i__4 = i__ - 1 + j * x_dim1; i__5 = i__; i__6 = i__ + 1 + j * x_dim1; rwork[i__] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(& bi), abs(d__2)) + ((d__3 = e[i__3].r, abs(d__3)) + (d__4 = d_imag(&e[i__ - 1]), abs(d__4))) * (( d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[ i__ - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8)) ) + ((d__9 = e[i__5].r, abs(d__9)) + (d__10 = d_imag(&e[i__]), abs(d__10))) * ((d__11 = x[i__6] .r, abs(d__11)) + (d__12 = d_imag(&x[i__ + 1 + j * x_dim1]), abs(d__12))); /* L30: */ } i__2 = *n + j * b_dim1; bi.r = b[i__2].r, bi.i = b[i__2].i; d_cnjg(&z__2, &e[*n - 1]); i__2 = *n - 1 + j * x_dim1; z__1.r = z__2.r * x[i__2].r - z__2.i * x[i__2].i, z__1.i = z__2.r * x[i__2].i + z__2.i * x[i__2].r; cx.r = z__1.r, cx.i = z__1.i; i__2 = *n; i__3 = *n + j * x_dim1; z__1.r = d__[i__2] * x[i__3].r, z__1.i = d__[i__2] * x[i__3] .i; dx.r = z__1.r, dx.i = z__1.i; i__2 = *n; z__2.r = bi.r - cx.r, z__2.i = bi.i - cx.i; z__1.r = z__2.r - dx.r, z__1.i = z__2.i - dx.i; work[i__2].r = z__1.r, work[i__2].i = z__1.i; i__2 = *n - 1; i__3 = *n - 1 + j * x_dim1; rwork[*n] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = e[i__2].r, abs(d__3)) + (d__4 = d_imag(&e[*n - 1]), abs(d__4))) * ((d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[*n - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8))); } } else { if (*n == 1) { i__2 = j * b_dim1 + 1; bi.r = b[i__2].r, bi.i = b[i__2].i; i__2 = j * x_dim1 + 1; z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i; dx.r = z__1.r, dx.i = z__1.i; z__1.r = bi.r - dx.r, z__1.i = bi.i - dx.i; work[1].r = z__1.r, work[1].i = z__1.i; rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 = d_imag(&dx), abs(d__4))); } else { i__2 = j * b_dim1 + 1; bi.r = b[i__2].r, bi.i = b[i__2].i; i__2 = j * x_dim1 + 1; z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i; dx.r = z__1.r, dx.i = z__1.i; d_cnjg(&z__2, &e[1]); i__2 = j * x_dim1 + 2; z__1.r = z__2.r * x[i__2].r - z__2.i * x[i__2].i, z__1.i = z__2.r * x[i__2].i + z__2.i * x[i__2].r; ex.r = z__1.r, ex.i = z__1.i; z__2.r = bi.r - dx.r, z__2.i = bi.i - dx.i; z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i; work[1].r = z__1.r, work[1].i = z__1.i; i__2 = j * x_dim1 + 2; rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 = d_imag(&dx), abs(d__4))) + ((d__5 = e[1].r, abs(d__5)) + (d__6 = d_imag(&e[1]), abs(d__6))) * ((d__7 = x[ i__2].r, abs(d__7)) + (d__8 = d_imag(&x[j * x_dim1 + 2]), abs(d__8))); i__2 = *n - 1; for (i__ = 2; i__ <= i__2; ++i__) { i__3 = i__ + j * b_dim1; bi.r = b[i__3].r, bi.i = b[i__3].i; i__3 = i__ - 1; i__4 = i__ - 1 + j * x_dim1; z__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i, z__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[ i__4].r; cx.r = z__1.r, cx.i = z__1.i; i__3 = i__; i__4 = i__ + j * x_dim1; z__1.r = d__[i__3] * x[i__4].r, z__1.i = d__[i__3] * x[ i__4].i; dx.r = z__1.r, dx.i = z__1.i; d_cnjg(&z__2, &e[i__]); i__3 = i__ + 1 + j * x_dim1; z__1.r = z__2.r * x[i__3].r - z__2.i * x[i__3].i, z__1.i = z__2.r * x[i__3].i + z__2.i * x[i__3].r; ex.r = z__1.r, ex.i = z__1.i; i__3 = i__; z__3.r = bi.r - cx.r, z__3.i = bi.i - cx.i; z__2.r = z__3.r - dx.r, z__2.i = z__3.i - dx.i; z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i; work[i__3].r = z__1.r, work[i__3].i = z__1.i; i__3 = i__ - 1; i__4 = i__ - 1 + j * x_dim1; i__5 = i__; i__6 = i__ + 1 + j * x_dim1; rwork[i__] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(& bi), abs(d__2)) + ((d__3 = e[i__3].r, abs(d__3)) + (d__4 = d_imag(&e[i__ - 1]), abs(d__4))) * (( d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[ i__ - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8)) ) + ((d__9 = e[i__5].r, abs(d__9)) + (d__10 = d_imag(&e[i__]), abs(d__10))) * ((d__11 = x[i__6] .r, abs(d__11)) + (d__12 = d_imag(&x[i__ + 1 + j * x_dim1]), abs(d__12))); /* L40: */ } i__2 = *n + j * b_dim1; bi.r = b[i__2].r, bi.i = b[i__2].i; i__2 = *n - 1; i__3 = *n - 1 + j * x_dim1; z__1.r = e[i__2].r * x[i__3].r - e[i__2].i * x[i__3].i, z__1.i = e[i__2].r * x[i__3].i + e[i__2].i * x[i__3] .r; cx.r = z__1.r, cx.i = z__1.i; i__2 = *n; i__3 = *n + j * x_dim1; z__1.r = d__[i__2] * x[i__3].r, z__1.i = d__[i__2] * x[i__3] .i; dx.r = z__1.r, dx.i = z__1.i; i__2 = *n; z__2.r = bi.r - cx.r, z__2.i = bi.i - cx.i; z__1.r = z__2.r - dx.r, z__1.i = z__2.i - dx.i; work[i__2].r = z__1.r, work[i__2].i = z__1.i; i__2 = *n - 1; i__3 = *n - 1 + j * x_dim1; rwork[*n] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi), abs(d__2)) + ((d__3 = e[i__2].r, abs(d__3)) + (d__4 = d_imag(&e[*n - 1]), abs(d__4))) * ((d__5 = x[i__3].r, abs(d__5)) + (d__6 = d_imag(&x[*n - 1 + j * x_dim1]), abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8))); } } /* Compute componentwise relative backward error from formula */ /* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */ /* where abs(Z) is the componentwise absolute value of the matrix */ /* or vector Z. If the i-th component of the denominator is less */ /* than SAFE2, then SAFE1 is added to the i-th components of the */ /* numerator and denominator before dividing. */ s = 0.; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { if (rwork[i__] > safe2) { /* Computing MAX */ i__3 = i__; d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2))) / rwork[i__]; s = max(d__3,d__4); } else { /* Computing MAX */ i__3 = i__; d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__] + safe1); s = max(d__3,d__4); } /* L50: */ } berr[j] = s; /* Test stopping criterion. Continue iterating if */ /* 1) The residual BERR(J) is larger than machine epsilon, and */ /* 2) BERR(J) decreased by at least a factor of 2 during the */ /* last iteration, and */ /* 3) At most ITMAX iterations tried. */ if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) { /* Update solution and try again. */ zpttrs_(uplo, n, &c__1, &df[1], &ef[1], &work[1], n, info); zaxpy_(n, &c_b16, &work[1], &c__1, &x[j * x_dim1 + 1], &c__1); lstres = berr[j]; ++count; goto L20; } /* Bound error from formula */ /* norm(X - XTRUE) / norm(X) .le. FERR = */ /* norm( abs(inv(A))* */ /* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */ /* where */ /* norm(Z) is the magnitude of the largest component of Z */ /* inv(A) is the inverse of A */ /* abs(Z) is the componentwise absolute value of the matrix or */ /* vector Z */ /* NZ is the maximum number of nonzeros in any row of A, plus 1 */ /* EPS is machine epsilon */ /* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */ /* is incremented by SAFE1 if the i-th component of */ /* abs(A)*abs(X) + abs(B) is less than SAFE2. */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { if (rwork[i__] > safe2) { i__3 = i__; rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__] ; } else { i__3 = i__; rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 = d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__] + safe1; } /* L60: */ } ix = idamax_(n, &rwork[1], &c__1); ferr[j] = rwork[ix]; /* Estimate the norm of inv(A). */ /* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by */ /* m(i,j) = abs(A(i,j)), i = j, */ /* m(i,j) = -abs(A(i,j)), i .ne. j, */ /* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. */ /* Solve M(L) * x = e. */ rwork[1] = 1.; i__2 = *n; for (i__ = 2; i__ <= i__2; ++i__) { rwork[i__] = rwork[i__ - 1] * z_abs(&ef[i__ - 1]) + 1.; /* L70: */ } /* Solve D * M(L)' * x = b. */ rwork[*n] /= df[*n]; for (i__ = *n - 1; i__ >= 1; --i__) { rwork[i__] = rwork[i__] / df[i__] + rwork[i__ + 1] * z_abs(&ef[ i__]); /* L80: */ } /* Compute norm(inv(A)) = max(x(i)), 1<=i<=n. */ ix = idamax_(n, &rwork[1], &c__1); ferr[j] *= (d__1 = rwork[ix], abs(d__1)); /* Normalize error. */ lstres = 0.; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { /* Computing MAX */ d__1 = lstres, d__2 = z_abs(&x[i__ + j * x_dim1]); lstres = max(d__1,d__2); /* L90: */ } if (lstres != 0.) { ferr[j] /= lstres; } /* L100: */ } return 0; /* End of ZPTRFS */ } /* zptrfs_ */