#include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static complex c_b16 = {1.f,0.f}; /* Subroutine */ int cptrfs_(char *uplo, integer *n, integer *nrhs, real *d__, complex *e, real *df, complex *ef, complex *b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr, complex *work, real *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; real r__1, r__2, r__3, r__4, r__5, r__6, r__7, r__8, r__9, r__10, r__11, r__12; complex q__1, q__2, q__3; /* Builtin functions */ double r_imag(complex *); void r_cnjg(complex *, complex *); double c_abs(complex *); /* Local variables */ integer i__, j; real s; complex bi, cx, dx, ex; integer ix, nz; real eps, safe1, safe2; extern logical lsame_(char *, char *); extern /* Subroutine */ int caxpy_(integer *, complex *, complex *, integer *, complex *, integer *); integer count; logical upper; extern doublereal slamch_(char *); real safmin; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer isamax_(integer *, real *, integer *); real lstres; extern /* Subroutine */ int cpttrs_(char *, integer *, integer *, real *, complex *, complex *, integer *, integer *); /* -- LAPACK routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* CPTRFS 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) REAL array, dimension (N) */ /* The n real diagonal elements of the tridiagonal matrix A. */ /* E (input) COMPLEX array, dimension (N-1) */ /* The (n-1) off-diagonal elements of the tridiagonal matrix A */ /* (see UPLO). */ /* DF (input) REAL array, dimension (N) */ /* The n diagonal elements of the diagonal matrix D from */ /* the factorization computed by CPTTRF. */ /* EF (input) COMPLEX array, dimension (N-1) */ /* The (n-1) off-diagonal elements of the unit bidiagonal */ /* factor U or L from the factorization computed by CPTTRF */ /* (see UPLO). */ /* B (input) COMPLEX 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 array, dimension (LDX,NRHS) */ /* On entry, the solution matrix X, as computed by CPTTRS. */ /* On exit, the improved solution matrix X. */ /* LDX (input) INTEGER */ /* The leading dimension of the array X. LDX >= max(1,N). */ /* FERR (output) REAL 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) REAL 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 array, dimension (N) */ /* RWORK (workspace) REAL 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_("CPTRFS", &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.f; berr[j] = 0.f; /* L10: */ } return 0; } /* NZ = maximum number of nonzero elements in each row of A, plus 1 */ nz = 4; eps = slamch_("Epsilon"); safmin = slamch_("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.f; 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; q__1.r = d__[1] * x[i__2].r, q__1.i = d__[1] * x[i__2].i; dx.r = q__1.r, dx.i = q__1.i; q__1.r = bi.r - dx.r, q__1.i = bi.i - dx.i; work[1].r = q__1.r, work[1].i = q__1.i; rwork[1] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = dx.r, dabs(r__3)) + (r__4 = r_imag(&dx), dabs(r__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; q__1.r = d__[1] * x[i__2].r, q__1.i = d__[1] * x[i__2].i; dx.r = q__1.r, dx.i = q__1.i; i__2 = j * x_dim1 + 2; q__1.r = e[1].r * x[i__2].r - e[1].i * x[i__2].i, q__1.i = e[ 1].r * x[i__2].i + e[1].i * x[i__2].r; ex.r = q__1.r, ex.i = q__1.i; q__2.r = bi.r - dx.r, q__2.i = bi.i - dx.i; q__1.r = q__2.r - ex.r, q__1.i = q__2.i - ex.i; work[1].r = q__1.r, work[1].i = q__1.i; i__2 = j * x_dim1 + 2; rwork[1] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = dx.r, dabs(r__3)) + (r__4 = r_imag(&dx), dabs(r__4))) + ((r__5 = e[1].r, dabs( r__5)) + (r__6 = r_imag(&e[1]), dabs(r__6))) * ((r__7 = x[i__2].r, dabs(r__7)) + (r__8 = r_imag(&x[j * x_dim1 + 2]), dabs(r__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; r_cnjg(&q__2, &e[i__ - 1]); i__3 = i__ - 1 + j * x_dim1; q__1.r = q__2.r * x[i__3].r - q__2.i * x[i__3].i, q__1.i = q__2.r * x[i__3].i + q__2.i * x[i__3].r; cx.r = q__1.r, cx.i = q__1.i; i__3 = i__; i__4 = i__ + j * x_dim1; q__1.r = d__[i__3] * x[i__4].r, q__1.i = d__[i__3] * x[ i__4].i; dx.r = q__1.r, dx.i = q__1.i; i__3 = i__; i__4 = i__ + 1 + j * x_dim1; q__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i, q__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[ i__4].r; ex.r = q__1.r, ex.i = q__1.i; i__3 = i__; q__3.r = bi.r - cx.r, q__3.i = bi.i - cx.i; q__2.r = q__3.r - dx.r, q__2.i = q__3.i - dx.i; q__1.r = q__2.r - ex.r, q__1.i = q__2.i - ex.i; work[i__3].r = q__1.r, work[i__3].i = q__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__] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(& bi), dabs(r__2)) + ((r__3 = e[i__3].r, dabs(r__3)) + (r__4 = r_imag(&e[i__ - 1]), dabs(r__4))) * (( r__5 = x[i__4].r, dabs(r__5)) + (r__6 = r_imag(&x[ i__ - 1 + j * x_dim1]), dabs(r__6))) + ((r__7 = dx.r, dabs(r__7)) + (r__8 = r_imag(&dx), dabs( r__8))) + ((r__9 = e[i__5].r, dabs(r__9)) + ( r__10 = r_imag(&e[i__]), dabs(r__10))) * ((r__11 = x[i__6].r, dabs(r__11)) + (r__12 = r_imag(&x[i__ + 1 + j * x_dim1]), dabs(r__12))); /* L30: */ } i__2 = *n + j * b_dim1; bi.r = b[i__2].r, bi.i = b[i__2].i; r_cnjg(&q__2, &e[*n - 1]); i__2 = *n - 1 + j * x_dim1; q__1.r = q__2.r * x[i__2].r - q__2.i * x[i__2].i, q__1.i = q__2.r * x[i__2].i + q__2.i * x[i__2].r; cx.r = q__1.r, cx.i = q__1.i; i__2 = *n; i__3 = *n + j * x_dim1; q__1.r = d__[i__2] * x[i__3].r, q__1.i = d__[i__2] * x[i__3] .i; dx.r = q__1.r, dx.i = q__1.i; i__2 = *n; q__2.r = bi.r - cx.r, q__2.i = bi.i - cx.i; q__1.r = q__2.r - dx.r, q__1.i = q__2.i - dx.i; work[i__2].r = q__1.r, work[i__2].i = q__1.i; i__2 = *n - 1; i__3 = *n - 1 + j * x_dim1; rwork[*n] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = e[i__2].r, dabs(r__3)) + (r__4 = r_imag(&e[*n - 1]), dabs(r__4))) * ((r__5 = x[i__3] .r, dabs(r__5)) + (r__6 = r_imag(&x[*n - 1 + j * x_dim1]), dabs(r__6))) + ((r__7 = dx.r, dabs(r__7)) + (r__8 = r_imag(&dx), dabs(r__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; q__1.r = d__[1] * x[i__2].r, q__1.i = d__[1] * x[i__2].i; dx.r = q__1.r, dx.i = q__1.i; q__1.r = bi.r - dx.r, q__1.i = bi.i - dx.i; work[1].r = q__1.r, work[1].i = q__1.i; rwork[1] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = dx.r, dabs(r__3)) + (r__4 = r_imag(&dx), dabs(r__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; q__1.r = d__[1] * x[i__2].r, q__1.i = d__[1] * x[i__2].i; dx.r = q__1.r, dx.i = q__1.i; r_cnjg(&q__2, &e[1]); i__2 = j * x_dim1 + 2; q__1.r = q__2.r * x[i__2].r - q__2.i * x[i__2].i, q__1.i = q__2.r * x[i__2].i + q__2.i * x[i__2].r; ex.r = q__1.r, ex.i = q__1.i; q__2.r = bi.r - dx.r, q__2.i = bi.i - dx.i; q__1.r = q__2.r - ex.r, q__1.i = q__2.i - ex.i; work[1].r = q__1.r, work[1].i = q__1.i; i__2 = j * x_dim1 + 2; rwork[1] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = dx.r, dabs(r__3)) + (r__4 = r_imag(&dx), dabs(r__4))) + ((r__5 = e[1].r, dabs( r__5)) + (r__6 = r_imag(&e[1]), dabs(r__6))) * ((r__7 = x[i__2].r, dabs(r__7)) + (r__8 = r_imag(&x[j * x_dim1 + 2]), dabs(r__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; q__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i, q__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[ i__4].r; cx.r = q__1.r, cx.i = q__1.i; i__3 = i__; i__4 = i__ + j * x_dim1; q__1.r = d__[i__3] * x[i__4].r, q__1.i = d__[i__3] * x[ i__4].i; dx.r = q__1.r, dx.i = q__1.i; r_cnjg(&q__2, &e[i__]); i__3 = i__ + 1 + j * x_dim1; q__1.r = q__2.r * x[i__3].r - q__2.i * x[i__3].i, q__1.i = q__2.r * x[i__3].i + q__2.i * x[i__3].r; ex.r = q__1.r, ex.i = q__1.i; i__3 = i__; q__3.r = bi.r - cx.r, q__3.i = bi.i - cx.i; q__2.r = q__3.r - dx.r, q__2.i = q__3.i - dx.i; q__1.r = q__2.r - ex.r, q__1.i = q__2.i - ex.i; work[i__3].r = q__1.r, work[i__3].i = q__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__] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(& bi), dabs(r__2)) + ((r__3 = e[i__3].r, dabs(r__3)) + (r__4 = r_imag(&e[i__ - 1]), dabs(r__4))) * (( r__5 = x[i__4].r, dabs(r__5)) + (r__6 = r_imag(&x[ i__ - 1 + j * x_dim1]), dabs(r__6))) + ((r__7 = dx.r, dabs(r__7)) + (r__8 = r_imag(&dx), dabs( r__8))) + ((r__9 = e[i__5].r, dabs(r__9)) + ( r__10 = r_imag(&e[i__]), dabs(r__10))) * ((r__11 = x[i__6].r, dabs(r__11)) + (r__12 = r_imag(&x[i__ + 1 + j * x_dim1]), dabs(r__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; q__1.r = e[i__2].r * x[i__3].r - e[i__2].i * x[i__3].i, q__1.i = e[i__2].r * x[i__3].i + e[i__2].i * x[i__3] .r; cx.r = q__1.r, cx.i = q__1.i; i__2 = *n; i__3 = *n + j * x_dim1; q__1.r = d__[i__2] * x[i__3].r, q__1.i = d__[i__2] * x[i__3] .i; dx.r = q__1.r, dx.i = q__1.i; i__2 = *n; q__2.r = bi.r - cx.r, q__2.i = bi.i - cx.i; q__1.r = q__2.r - dx.r, q__1.i = q__2.i - dx.i; work[i__2].r = q__1.r, work[i__2].i = q__1.i; i__2 = *n - 1; i__3 = *n - 1 + j * x_dim1; rwork[*n] = (r__1 = bi.r, dabs(r__1)) + (r__2 = r_imag(&bi), dabs(r__2)) + ((r__3 = e[i__2].r, dabs(r__3)) + (r__4 = r_imag(&e[*n - 1]), dabs(r__4))) * ((r__5 = x[i__3] .r, dabs(r__5)) + (r__6 = r_imag(&x[*n - 1 + j * x_dim1]), dabs(r__6))) + ((r__7 = dx.r, dabs(r__7)) + (r__8 = r_imag(&dx), dabs(r__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.f; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { if (rwork[i__] > safe2) { /* Computing MAX */ i__3 = i__; r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2))) / rwork[i__]; s = dmax(r__3,r__4); } else { /* Computing MAX */ i__3 = i__; r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__] + safe1); s = dmax(r__3,r__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.f <= lstres && count <= 5) { /* Update solution and try again. */ cpttrs_(uplo, n, &c__1, &df[1], &ef[1], &work[1], n, info); caxpy_(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__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[ i__]; } else { i__3 = i__; rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[ i__] + safe1; } /* L60: */ } ix = isamax_(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.f; i__2 = *n; for (i__ = 2; i__ <= i__2; ++i__) { rwork[i__] = rwork[i__ - 1] * c_abs(&ef[i__ - 1]) + 1.f; /* 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] * c_abs(&ef[ i__]); /* L80: */ } /* Compute norm(inv(A)) = max(x(i)), 1<=i<=n. */ ix = isamax_(n, &rwork[1], &c__1); ferr[j] *= (r__1 = rwork[ix], dabs(r__1)); /* Normalize error. */ lstres = 0.f; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { /* Computing MAX */ r__1 = lstres, r__2 = c_abs(&x[i__ + j * x_dim1]); lstres = dmax(r__1,r__2); /* L90: */ } if (lstres != 0.f) { ferr[j] /= lstres; } /* L100: */ } return 0; /* End of CPTRFS */ } /* cptrfs_ */