#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int csytf2_(char *uplo, integer *n, complex *a, integer *lda, integer *ipiv, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U' is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS. Arguments ========= UPLO (input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) COMPLEX array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). IPIV (output) INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. Further Details =============== 09-29-06 - patch from Bobby Cheng, MathWorks Replace l.209 and l.377 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN by IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN 1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company If UPLO = 'U', then A = U*D*U', where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L', where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static complex c_b1 = {1.f,0.f}; static integer c__1 = 1; /* System generated locals */ integer a_dim1, a_offset, i__1, i__2, i__3, i__4, i__5, i__6; real r__1, r__2, r__3, r__4; complex q__1, q__2, q__3, q__4; /* Builtin functions */ double sqrt(doublereal), r_imag(complex *); void c_div(complex *, complex *, complex *); /* Local variables */ static integer i__, j, k; static complex t, r1, d11, d12, d21, d22; static integer kk, kp; static complex wk, wkm1, wkp1; static integer imax, jmax; extern /* Subroutine */ int csyr_(char *, integer *, complex *, complex *, integer *, complex *, integer *); static real alpha; extern /* Subroutine */ int cscal_(integer *, complex *, complex *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int cswap_(integer *, complex *, integer *, complex *, integer *); static integer kstep; static logical upper; static real absakk; extern integer icamax_(integer *, complex *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *); static real colmax; extern logical sisnan_(real *); static real rowmax; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --ipiv; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < max(1,*n)) { *info = -4; } if (*info != 0) { i__1 = -(*info); xerbla_("CSYTF2", &i__1); return 0; } /* Initialize ALPHA for use in choosing pivot block size. */ alpha = (sqrt(17.f) + 1.f) / 8.f; if (upper) { /* Factorize A as U*D*U' using the upper triangle of A K is the main loop index, decreasing from N to 1 in steps of 1 or 2 */ k = *n; L10: /* If K < 1, exit from loop */ if (k < 1) { goto L70; } kstep = 1; /* Determine rows and columns to be interchanged and whether a 1-by-1 or 2-by-2 pivot block will be used */ i__1 = k + k * a_dim1; absakk = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[k + k * a_dim1]), dabs(r__2)); /* IMAX is the row-index of the largest off-diagonal element in column K, and COLMAX is its absolute value */ if (k > 1) { i__1 = k - 1; imax = icamax_(&i__1, &a[k * a_dim1 + 1], &c__1); i__1 = imax + k * a_dim1; colmax = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[imax + k * a_dim1]), dabs(r__2)); } else { colmax = 0.f; } if (dmax(absakk,colmax) == 0.f || sisnan_(&absakk)) { /* Column K is zero or NaN: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; } else { if (absakk >= alpha * colmax) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { /* JMAX is the column-index of the largest off-diagonal element in row IMAX, and ROWMAX is its absolute value */ i__1 = k - imax; jmax = imax + icamax_(&i__1, &a[imax + (imax + 1) * a_dim1], lda); i__1 = imax + jmax * a_dim1; rowmax = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[ imax + jmax * a_dim1]), dabs(r__2)); if (imax > 1) { i__1 = imax - 1; jmax = icamax_(&i__1, &a[imax * a_dim1 + 1], &c__1); /* Computing MAX */ i__1 = jmax + imax * a_dim1; r__3 = rowmax, r__4 = (r__1 = a[i__1].r, dabs(r__1)) + ( r__2 = r_imag(&a[jmax + imax * a_dim1]), dabs( r__2)); rowmax = dmax(r__3,r__4); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else /* if(complicated condition) */ { i__1 = imax + imax * a_dim1; if ((r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[ imax + imax * a_dim1]), dabs(r__2)) >= alpha * rowmax) { /* interchange rows and columns K and IMAX, use 1-by-1 pivot block */ kp = imax; } else { /* interchange rows and columns K-1 and IMAX, use 2-by-2 pivot block */ kp = imax; kstep = 2; } } } kk = k - kstep + 1; if (kp != kk) { /* Interchange rows and columns KK and KP in the leading submatrix A(1:k,1:k) */ i__1 = kp - 1; cswap_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1], &c__1); i__1 = kk - kp - 1; cswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (kp + 1) * a_dim1], lda); i__1 = kk + kk * a_dim1; t.r = a[i__1].r, t.i = a[i__1].i; i__1 = kk + kk * a_dim1; i__2 = kp + kp * a_dim1; a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i; i__1 = kp + kp * a_dim1; a[i__1].r = t.r, a[i__1].i = t.i; if (kstep == 2) { i__1 = k - 1 + k * a_dim1; t.r = a[i__1].r, t.i = a[i__1].i; i__1 = k - 1 + k * a_dim1; i__2 = kp + k * a_dim1; a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i; i__1 = kp + k * a_dim1; a[i__1].r = t.r, a[i__1].i = t.i; } } /* Update the leading submatrix */ if (kstep == 1) { /* 1-by-1 pivot block D(k): column k now holds W(k) = U(k)*D(k) where U(k) is the k-th column of U Perform a rank-1 update of A(1:k-1,1:k-1) as A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' */ c_div(&q__1, &c_b1, &a[k + k * a_dim1]); r1.r = q__1.r, r1.i = q__1.i; i__1 = k - 1; q__1.r = -r1.r, q__1.i = -r1.i; csyr_(uplo, &i__1, &q__1, &a[k * a_dim1 + 1], &c__1, &a[ a_offset], lda); /* Store U(k) in column k */ i__1 = k - 1; cscal_(&i__1, &r1, &a[k * a_dim1 + 1], &c__1); } else { /* 2-by-2 pivot block D(k): columns k and k-1 now hold ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) where U(k) and U(k-1) are the k-th and (k-1)-th columns of U Perform a rank-2 update of A(1:k-2,1:k-2) as A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' */ if (k > 2) { i__1 = k - 1 + k * a_dim1; d12.r = a[i__1].r, d12.i = a[i__1].i; c_div(&q__1, &a[k - 1 + (k - 1) * a_dim1], &d12); d22.r = q__1.r, d22.i = q__1.i; c_div(&q__1, &a[k + k * a_dim1], &d12); d11.r = q__1.r, d11.i = q__1.i; q__3.r = d11.r * d22.r - d11.i * d22.i, q__3.i = d11.r * d22.i + d11.i * d22.r; q__2.r = q__3.r - 1.f, q__2.i = q__3.i - 0.f; c_div(&q__1, &c_b1, &q__2); t.r = q__1.r, t.i = q__1.i; c_div(&q__1, &t, &d12); d12.r = q__1.r, d12.i = q__1.i; for (j = k - 2; j >= 1; --j) { i__1 = j + (k - 1) * a_dim1; q__3.r = d11.r * a[i__1].r - d11.i * a[i__1].i, q__3.i = d11.r * a[i__1].i + d11.i * a[i__1] .r; i__2 = j + k * a_dim1; q__2.r = q__3.r - a[i__2].r, q__2.i = q__3.i - a[i__2] .i; q__1.r = d12.r * q__2.r - d12.i * q__2.i, q__1.i = d12.r * q__2.i + d12.i * q__2.r; wkm1.r = q__1.r, wkm1.i = q__1.i; i__1 = j + k * a_dim1; q__3.r = d22.r * a[i__1].r - d22.i * a[i__1].i, q__3.i = d22.r * a[i__1].i + d22.i * a[i__1] .r; i__2 = j + (k - 1) * a_dim1; q__2.r = q__3.r - a[i__2].r, q__2.i = q__3.i - a[i__2] .i; q__1.r = d12.r * q__2.r - d12.i * q__2.i, q__1.i = d12.r * q__2.i + d12.i * q__2.r; wk.r = q__1.r, wk.i = q__1.i; for (i__ = j; i__ >= 1; --i__) { i__1 = i__ + j * a_dim1; i__2 = i__ + j * a_dim1; i__3 = i__ + k * a_dim1; q__3.r = a[i__3].r * wk.r - a[i__3].i * wk.i, q__3.i = a[i__3].r * wk.i + a[i__3].i * wk.r; q__2.r = a[i__2].r - q__3.r, q__2.i = a[i__2].i - q__3.i; i__4 = i__ + (k - 1) * a_dim1; q__4.r = a[i__4].r * wkm1.r - a[i__4].i * wkm1.i, q__4.i = a[i__4].r * wkm1.i + a[i__4].i * wkm1.r; q__1.r = q__2.r - q__4.r, q__1.i = q__2.i - q__4.i; a[i__1].r = q__1.r, a[i__1].i = q__1.i; /* L20: */ } i__1 = j + k * a_dim1; a[i__1].r = wk.r, a[i__1].i = wk.i; i__1 = j + (k - 1) * a_dim1; a[i__1].r = wkm1.r, a[i__1].i = wkm1.i; /* L30: */ } } } } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -kp; ipiv[k - 1] = -kp; } /* Decrease K and return to the start of the main loop */ k -= kstep; goto L10; } else { /* Factorize A as L*D*L' using the lower triangle of A K is the main loop index, increasing from 1 to N in steps of 1 or 2 */ k = 1; L40: /* If K > N, exit from loop */ if (k > *n) { goto L70; } kstep = 1; /* Determine rows and columns to be interchanged and whether a 1-by-1 or 2-by-2 pivot block will be used */ i__1 = k + k * a_dim1; absakk = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[k + k * a_dim1]), dabs(r__2)); /* IMAX is the row-index of the largest off-diagonal element in column K, and COLMAX is its absolute value */ if (k < *n) { i__1 = *n - k; imax = k + icamax_(&i__1, &a[k + 1 + k * a_dim1], &c__1); i__1 = imax + k * a_dim1; colmax = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[imax + k * a_dim1]), dabs(r__2)); } else { colmax = 0.f; } if (dmax(absakk,colmax) == 0.f || sisnan_(&absakk)) { /* Column K is zero or NaN: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; } else { if (absakk >= alpha * colmax) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { /* JMAX is the column-index of the largest off-diagonal element in row IMAX, and ROWMAX is its absolute value */ i__1 = imax - k; jmax = k - 1 + icamax_(&i__1, &a[imax + k * a_dim1], lda); i__1 = imax + jmax * a_dim1; rowmax = (r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[ imax + jmax * a_dim1]), dabs(r__2)); if (imax < *n) { i__1 = *n - imax; jmax = imax + icamax_(&i__1, &a[imax + 1 + imax * a_dim1], &c__1); /* Computing MAX */ i__1 = jmax + imax * a_dim1; r__3 = rowmax, r__4 = (r__1 = a[i__1].r, dabs(r__1)) + ( r__2 = r_imag(&a[jmax + imax * a_dim1]), dabs( r__2)); rowmax = dmax(r__3,r__4); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else /* if(complicated condition) */ { i__1 = imax + imax * a_dim1; if ((r__1 = a[i__1].r, dabs(r__1)) + (r__2 = r_imag(&a[ imax + imax * a_dim1]), dabs(r__2)) >= alpha * rowmax) { /* interchange rows and columns K and IMAX, use 1-by-1 pivot block */ kp = imax; } else { /* interchange rows and columns K+1 and IMAX, use 2-by-2 pivot block */ kp = imax; kstep = 2; } } } kk = k + kstep - 1; if (kp != kk) { /* Interchange rows and columns KK and KP in the trailing submatrix A(k:n,k:n) */ if (kp < *n) { i__1 = *n - kp; cswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1 + kp * a_dim1], &c__1); } i__1 = kp - kk - 1; cswap_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + (kk + 1) * a_dim1], lda); i__1 = kk + kk * a_dim1; t.r = a[i__1].r, t.i = a[i__1].i; i__1 = kk + kk * a_dim1; i__2 = kp + kp * a_dim1; a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i; i__1 = kp + kp * a_dim1; a[i__1].r = t.r, a[i__1].i = t.i; if (kstep == 2) { i__1 = k + 1 + k * a_dim1; t.r = a[i__1].r, t.i = a[i__1].i; i__1 = k + 1 + k * a_dim1; i__2 = kp + k * a_dim1; a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i; i__1 = kp + k * a_dim1; a[i__1].r = t.r, a[i__1].i = t.i; } } /* Update the trailing submatrix */ if (kstep == 1) { /* 1-by-1 pivot block D(k): column k now holds W(k) = L(k)*D(k) where L(k) is the k-th column of L */ if (k < *n) { /* Perform a rank-1 update of A(k+1:n,k+1:n) as A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' */ c_div(&q__1, &c_b1, &a[k + k * a_dim1]); r1.r = q__1.r, r1.i = q__1.i; i__1 = *n - k; q__1.r = -r1.r, q__1.i = -r1.i; csyr_(uplo, &i__1, &q__1, &a[k + 1 + k * a_dim1], &c__1, & a[k + 1 + (k + 1) * a_dim1], lda); /* Store L(k) in column K */ i__1 = *n - k; cscal_(&i__1, &r1, &a[k + 1 + k * a_dim1], &c__1); } } else { /* 2-by-2 pivot block D(k) */ if (k < *n - 1) { /* Perform a rank-2 update of A(k+2:n,k+2:n) as A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' where L(k) and L(k+1) are the k-th and (k+1)-th columns of L */ i__1 = k + 1 + k * a_dim1; d21.r = a[i__1].r, d21.i = a[i__1].i; c_div(&q__1, &a[k + 1 + (k + 1) * a_dim1], &d21); d11.r = q__1.r, d11.i = q__1.i; c_div(&q__1, &a[k + k * a_dim1], &d21); d22.r = q__1.r, d22.i = q__1.i; q__3.r = d11.r * d22.r - d11.i * d22.i, q__3.i = d11.r * d22.i + d11.i * d22.r; q__2.r = q__3.r - 1.f, q__2.i = q__3.i - 0.f; c_div(&q__1, &c_b1, &q__2); t.r = q__1.r, t.i = q__1.i; c_div(&q__1, &t, &d21); d21.r = q__1.r, d21.i = q__1.i; i__1 = *n; for (j = k + 2; j <= i__1; ++j) { i__2 = j + k * a_dim1; q__3.r = d11.r * a[i__2].r - d11.i * a[i__2].i, q__3.i = d11.r * a[i__2].i + d11.i * a[i__2] .r; i__3 = j + (k + 1) * a_dim1; q__2.r = q__3.r - a[i__3].r, q__2.i = q__3.i - a[i__3] .i; q__1.r = d21.r * q__2.r - d21.i * q__2.i, q__1.i = d21.r * q__2.i + d21.i * q__2.r; wk.r = q__1.r, wk.i = q__1.i; i__2 = j + (k + 1) * a_dim1; q__3.r = d22.r * a[i__2].r - d22.i * a[i__2].i, q__3.i = d22.r * a[i__2].i + d22.i * a[i__2] .r; i__3 = j + k * a_dim1; q__2.r = q__3.r - a[i__3].r, q__2.i = q__3.i - a[i__3] .i; q__1.r = d21.r * q__2.r - d21.i * q__2.i, q__1.i = d21.r * q__2.i + d21.i * q__2.r; wkp1.r = q__1.r, wkp1.i = q__1.i; i__2 = *n; for (i__ = j; i__ <= i__2; ++i__) { i__3 = i__ + j * a_dim1; i__4 = i__ + j * a_dim1; i__5 = i__ + k * a_dim1; q__3.r = a[i__5].r * wk.r - a[i__5].i * wk.i, q__3.i = a[i__5].r * wk.i + a[i__5].i * wk.r; q__2.r = a[i__4].r - q__3.r, q__2.i = a[i__4].i - q__3.i; i__6 = i__ + (k + 1) * a_dim1; q__4.r = a[i__6].r * wkp1.r - a[i__6].i * wkp1.i, q__4.i = a[i__6].r * wkp1.i + a[i__6].i * wkp1.r; q__1.r = q__2.r - q__4.r, q__1.i = q__2.i - q__4.i; a[i__3].r = q__1.r, a[i__3].i = q__1.i; /* L50: */ } i__2 = j + k * a_dim1; a[i__2].r = wk.r, a[i__2].i = wk.i; i__2 = j + (k + 1) * a_dim1; a[i__2].r = wkp1.r, a[i__2].i = wkp1.i; /* L60: */ } } } } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -kp; ipiv[k + 1] = -kp; } /* Increase K and return to the start of the main loop */ k += kstep; goto L40; } L70: return 0; /* End of CSYTF2 */ } /* csytf2_ */