#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int csptrf_(char *uplo, integer *n, complex *ap, integer * ipiv, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= CSPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. Arguments ========= UPLO (input) CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) INTEGER The order of the matrix A. N >= 0. AP (input/output) COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details). 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 = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) 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 =============== 5-96 - 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 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 imax, jmax; extern /* Subroutine */ int cspr_(char *, integer *, complex *, complex *, integer *, complex *); static integer i__, j, k; static complex t; 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 complex r1, d11, d12, d21, d22; static integer kc, kk, kp; static real absakk; static complex wk; static integer kx; extern integer icamax_(integer *, complex *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *); static real colmax, rowmax; static integer knc, kpc, npp; static complex wkm1, wkp1; --ipiv; --ap; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } if (*info != 0) { i__1 = -(*info); xerbla_("CSPTRF", &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; kc = (*n - 1) * *n / 2 + 1; L10: knc = kc; /* If K < 1, exit from loop */ if (k < 1) { goto L110; } 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 = kc + k - 1; absakk = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + k - 1]), 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, &ap[kc], &c__1); i__1 = kc + imax - 1; colmax = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + imax - 1]), dabs(r__2)); } else { colmax = 0.f; } if (dmax(absakk,colmax) == 0.f) { /* Column K is zero: 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 */ rowmax = 0.f; jmax = imax; kx = imax * (imax + 1) / 2 + imax; i__1 = k; for (j = imax + 1; j <= i__1; ++j) { i__2 = kx; if ((r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[ kx]), dabs(r__2)) > rowmax) { i__2 = kx; rowmax = (r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[kx]), dabs(r__2)); jmax = j; } kx += j; /* L20: */ } kpc = (imax - 1) * imax / 2 + 1; if (imax > 1) { i__1 = imax - 1; jmax = icamax_(&i__1, &ap[kpc], &c__1); /* Computing MAX */ i__1 = kpc + jmax - 1; r__3 = rowmax, r__4 = (r__1 = ap[i__1].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kpc + jmax - 1]), 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 = kpc + imax - 1; if ((r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[ kpc + imax - 1]), 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 (kstep == 2) { knc = knc - k + 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, &ap[knc], &c__1, &ap[kpc], &c__1); kx = kpc + kp - 1; i__1 = kk - 1; for (j = kp + 1; j <= i__1; ++j) { kx = kx + j - 1; i__2 = knc + j - 1; t.r = ap[i__2].r, t.i = ap[i__2].i; i__2 = knc + j - 1; i__3 = kx; ap[i__2].r = ap[i__3].r, ap[i__2].i = ap[i__3].i; i__2 = kx; ap[i__2].r = t.r, ap[i__2].i = t.i; /* L30: */ } i__1 = knc + kk - 1; t.r = ap[i__1].r, t.i = ap[i__1].i; i__1 = knc + kk - 1; i__2 = kpc + kp - 1; ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i; i__1 = kpc + kp - 1; ap[i__1].r = t.r, ap[i__1].i = t.i; if (kstep == 2) { i__1 = kc + k - 2; t.r = ap[i__1].r, t.i = ap[i__1].i; i__1 = kc + k - 2; i__2 = kc + kp - 1; ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i; i__1 = kc + kp - 1; ap[i__1].r = t.r, ap[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, &ap[kc + k - 1]); r1.r = q__1.r, r1.i = q__1.i; i__1 = k - 1; q__1.r = -r1.r, q__1.i = -r1.i; cspr_(uplo, &i__1, &q__1, &ap[kc], &c__1, &ap[1]); /* Store U(k) in column k */ i__1 = k - 1; cscal_(&i__1, &r1, &ap[kc], &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 - 1) * k / 2; d12.r = ap[i__1].r, d12.i = ap[i__1].i; c_div(&q__1, &ap[k - 1 + (k - 2) * (k - 1) / 2], &d12); d22.r = q__1.r, d22.i = q__1.i; c_div(&q__1, &ap[k + (k - 1) * k / 2], &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 - 2) * (k - 1) / 2; q__3.r = d11.r * ap[i__1].r - d11.i * ap[i__1].i, q__3.i = d11.r * ap[i__1].i + d11.i * ap[i__1] .r; i__2 = j + (k - 1) * k / 2; q__2.r = q__3.r - ap[i__2].r, q__2.i = q__3.i - ap[ 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 - 1) * k / 2; q__3.r = d22.r * ap[i__1].r - d22.i * ap[i__1].i, q__3.i = d22.r * ap[i__1].i + d22.i * ap[i__1] .r; i__2 = j + (k - 2) * (k - 1) / 2; q__2.r = q__3.r - ap[i__2].r, q__2.i = q__3.i - ap[ 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 - 1) * j / 2; i__2 = i__ + (j - 1) * j / 2; i__3 = i__ + (k - 1) * k / 2; q__3.r = ap[i__3].r * wk.r - ap[i__3].i * wk.i, q__3.i = ap[i__3].r * wk.i + ap[i__3].i * wk.r; q__2.r = ap[i__2].r - q__3.r, q__2.i = ap[i__2].i - q__3.i; i__4 = i__ + (k - 2) * (k - 1) / 2; q__4.r = ap[i__4].r * wkm1.r - ap[i__4].i * wkm1.i, q__4.i = ap[i__4].r * wkm1.i + ap[ i__4].i * wkm1.r; q__1.r = q__2.r - q__4.r, q__1.i = q__2.i - q__4.i; ap[i__1].r = q__1.r, ap[i__1].i = q__1.i; /* L40: */ } i__1 = j + (k - 1) * k / 2; ap[i__1].r = wk.r, ap[i__1].i = wk.i; i__1 = j + (k - 2) * (k - 1) / 2; ap[i__1].r = wkm1.r, ap[i__1].i = wkm1.i; /* L50: */ } } } } /* 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; kc = knc - k; 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; kc = 1; npp = *n * (*n + 1) / 2; L60: knc = kc; /* If K > N, exit from loop */ if (k > *n) { goto L110; } 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 = kc; absakk = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc]), 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, &ap[kc + 1], &c__1); i__1 = kc + imax - k; colmax = (r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + imax - k]), dabs(r__2)); } else { colmax = 0.f; } if (dmax(absakk,colmax) == 0.f) { /* Column K is zero: 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 */ rowmax = 0.f; kx = kc + imax - k; i__1 = imax - 1; for (j = k; j <= i__1; ++j) { i__2 = kx; if ((r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[ kx]), dabs(r__2)) > rowmax) { i__2 = kx; rowmax = (r__1 = ap[i__2].r, dabs(r__1)) + (r__2 = r_imag(&ap[kx]), dabs(r__2)); jmax = j; } kx = kx + *n - j; /* L70: */ } kpc = npp - (*n - imax + 1) * (*n - imax + 2) / 2 + 1; if (imax < *n) { i__1 = *n - imax; jmax = imax + icamax_(&i__1, &ap[kpc + 1], &c__1); /* Computing MAX */ i__1 = kpc + jmax - imax; r__3 = rowmax, r__4 = (r__1 = ap[i__1].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kpc + jmax - imax]), 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 = kpc; if ((r__1 = ap[i__1].r, dabs(r__1)) + (r__2 = r_imag(&ap[ kpc]), 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 (kstep == 2) { knc = knc + *n - k + 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, &ap[knc + kp - kk + 1], &c__1, &ap[kpc + 1], &c__1); } kx = knc + kp - kk; i__1 = kp - 1; for (j = kk + 1; j <= i__1; ++j) { kx = kx + *n - j + 1; i__2 = knc + j - kk; t.r = ap[i__2].r, t.i = ap[i__2].i; i__2 = knc + j - kk; i__3 = kx; ap[i__2].r = ap[i__3].r, ap[i__2].i = ap[i__3].i; i__2 = kx; ap[i__2].r = t.r, ap[i__2].i = t.i; /* L80: */ } i__1 = knc; t.r = ap[i__1].r, t.i = ap[i__1].i; i__1 = knc; i__2 = kpc; ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i; i__1 = kpc; ap[i__1].r = t.r, ap[i__1].i = t.i; if (kstep == 2) { i__1 = kc + 1; t.r = ap[i__1].r, t.i = ap[i__1].i; i__1 = kc + 1; i__2 = kc + kp - k; ap[i__1].r = ap[i__2].r, ap[i__1].i = ap[i__2].i; i__1 = kc + kp - k; ap[i__1].r = t.r, ap[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, &ap[kc]); r1.r = q__1.r, r1.i = q__1.i; i__1 = *n - k; q__1.r = -r1.r, q__1.i = -r1.i; cspr_(uplo, &i__1, &q__1, &ap[kc + 1], &c__1, &ap[kc + *n - k + 1]); /* Store L(k) in column K */ i__1 = *n - k; cscal_(&i__1, &r1, &ap[kc + 1], &c__1); } } else { /* 2-by-2 pivot block D(k): columns K and K+1 now hold ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) where L(k) and L(k+1) are the k-th and (k+1)-th columns of L */ 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 - 1) * ((*n << 1) - k) / 2; d21.r = ap[i__1].r, d21.i = ap[i__1].i; c_div(&q__1, &ap[k + 1 + k * ((*n << 1) - k - 1) / 2], & d21); d11.r = q__1.r, d11.i = q__1.i; c_div(&q__1, &ap[k + (k - 1) * ((*n << 1) - k) / 2], &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 - 1) * ((*n << 1) - k) / 2; q__3.r = d11.r * ap[i__2].r - d11.i * ap[i__2].i, q__3.i = d11.r * ap[i__2].i + d11.i * ap[i__2] .r; i__3 = j + k * ((*n << 1) - k - 1) / 2; q__2.r = q__3.r - ap[i__3].r, q__2.i = q__3.i - ap[ 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 * ((*n << 1) - k - 1) / 2; q__3.r = d22.r * ap[i__2].r - d22.i * ap[i__2].i, q__3.i = d22.r * ap[i__2].i + d22.i * ap[i__2] .r; i__3 = j + (k - 1) * ((*n << 1) - k) / 2; q__2.r = q__3.r - ap[i__3].r, q__2.i = q__3.i - ap[ 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 - 1) * ((*n << 1) - j) / 2; i__4 = i__ + (j - 1) * ((*n << 1) - j) / 2; i__5 = i__ + (k - 1) * ((*n << 1) - k) / 2; q__3.r = ap[i__5].r * wk.r - ap[i__5].i * wk.i, q__3.i = ap[i__5].r * wk.i + ap[i__5].i * wk.r; q__2.r = ap[i__4].r - q__3.r, q__2.i = ap[i__4].i - q__3.i; i__6 = i__ + k * ((*n << 1) - k - 1) / 2; q__4.r = ap[i__6].r * wkp1.r - ap[i__6].i * wkp1.i, q__4.i = ap[i__6].r * wkp1.i + ap[ i__6].i * wkp1.r; q__1.r = q__2.r - q__4.r, q__1.i = q__2.i - q__4.i; ap[i__3].r = q__1.r, ap[i__3].i = q__1.i; /* L90: */ } i__2 = j + (k - 1) * ((*n << 1) - k) / 2; ap[i__2].r = wk.r, ap[i__2].i = wk.i; i__2 = j + k * ((*n << 1) - k - 1) / 2; ap[i__2].r = wkp1.r, ap[i__2].i = wkp1.i; /* L100: */ } } } } /* 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; kc = knc + *n - k + 2; goto L60; } L110: return 0; /* End of CSPTRF */ } /* csptrf_ */