#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int ssptrf_(char *uplo, integer *n, real *ap, integer *ipiv, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SSPTRF computes the factorization of a real 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) REAL 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 integer c__1 = 1; /* System generated locals */ integer i__1, i__2; real r__1, r__2, r__3; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, k; static real t, r1, d11, d12, d21, d22; static integer kc, kk, kp; static real wk; static integer kx, knc, kpc, npp; static real wkm1, wkp1; static integer imax, jmax; extern /* Subroutine */ int sspr_(char *, integer *, real *, real *, integer *, real *); static real alpha; extern logical lsame_(char *, char *); extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); static integer kstep; static logical upper; extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, integer *); static real absakk; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer isamax_(integer *, real *, integer *); static real colmax, rowmax; --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_("SSPTRF", &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 */ absakk = (r__1 = ap[kc + k - 1], dabs(r__1)); /* 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 = isamax_(&i__1, &ap[kc], &c__1); colmax = (r__1 = ap[kc + imax - 1], dabs(r__1)); } 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) { if ((r__1 = ap[kx], dabs(r__1)) > rowmax) { rowmax = (r__1 = ap[kx], dabs(r__1)); jmax = j; } kx += j; /* L20: */ } kpc = (imax - 1) * imax / 2 + 1; if (imax > 1) { i__1 = imax - 1; jmax = isamax_(&i__1, &ap[kpc], &c__1); /* Computing MAX */ r__2 = rowmax, r__3 = (r__1 = ap[kpc + jmax - 1], dabs( r__1)); rowmax = dmax(r__2,r__3); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else if ((r__1 = ap[kpc + imax - 1], dabs(r__1)) >= 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; sswap_(&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; t = ap[knc + j - 1]; ap[knc + j - 1] = ap[kx]; ap[kx] = t; /* L30: */ } t = ap[knc + kk - 1]; ap[knc + kk - 1] = ap[kpc + kp - 1]; ap[kpc + kp - 1] = t; if (kstep == 2) { t = ap[kc + k - 2]; ap[kc + k - 2] = ap[kc + kp - 1]; ap[kc + kp - 1] = t; } } /* 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)' */ r1 = 1.f / ap[kc + k - 1]; i__1 = k - 1; r__1 = -r1; sspr_(uplo, &i__1, &r__1, &ap[kc], &c__1, &ap[1]); /* Store U(k) in column k */ i__1 = k - 1; sscal_(&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) { d12 = ap[k - 1 + (k - 1) * k / 2]; d22 = ap[k - 1 + (k - 2) * (k - 1) / 2] / d12; d11 = ap[k + (k - 1) * k / 2] / d12; t = 1.f / (d11 * d22 - 1.f); d12 = t / d12; for (j = k - 2; j >= 1; --j) { wkm1 = d12 * (d11 * ap[j + (k - 2) * (k - 1) / 2] - ap[j + (k - 1) * k / 2]); wk = d12 * (d22 * ap[j + (k - 1) * k / 2] - ap[j + (k - 2) * (k - 1) / 2]); for (i__ = j; i__ >= 1; --i__) { ap[i__ + (j - 1) * j / 2] = ap[i__ + (j - 1) * j / 2] - ap[i__ + (k - 1) * k / 2] * wk - ap[ i__ + (k - 2) * (k - 1) / 2] * wkm1; /* L40: */ } ap[j + (k - 1) * k / 2] = wk; ap[j + (k - 2) * (k - 1) / 2] = wkm1; /* 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 */ absakk = (r__1 = ap[kc], dabs(r__1)); /* 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 + isamax_(&i__1, &ap[kc + 1], &c__1); colmax = (r__1 = ap[kc + imax - k], dabs(r__1)); } 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) { if ((r__1 = ap[kx], dabs(r__1)) > rowmax) { rowmax = (r__1 = ap[kx], dabs(r__1)); 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 + isamax_(&i__1, &ap[kpc + 1], &c__1); /* Computing MAX */ r__2 = rowmax, r__3 = (r__1 = ap[kpc + jmax - imax], dabs( r__1)); rowmax = dmax(r__2,r__3); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else if ((r__1 = ap[kpc], dabs(r__1)) >= 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; sswap_(&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; t = ap[knc + j - kk]; ap[knc + j - kk] = ap[kx]; ap[kx] = t; /* L80: */ } t = ap[knc]; ap[knc] = ap[kpc]; ap[kpc] = t; if (kstep == 2) { t = ap[kc + 1]; ap[kc + 1] = ap[kc + kp - k]; ap[kc + kp - k] = t; } } /* 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)' */ r1 = 1.f / ap[kc]; i__1 = *n - k; r__1 = -r1; sspr_(uplo, &i__1, &r__1, &ap[kc + 1], &c__1, &ap[kc + *n - k + 1]); /* Store L(k) in column K */ i__1 = *n - k; sscal_(&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) )' */ d21 = ap[k + 1 + (k - 1) * ((*n << 1) - k) / 2]; d11 = ap[k + 1 + k * ((*n << 1) - k - 1) / 2] / d21; d22 = ap[k + (k - 1) * ((*n << 1) - k) / 2] / d21; t = 1.f / (d11 * d22 - 1.f); d21 = t / d21; i__1 = *n; for (j = k + 2; j <= i__1; ++j) { wk = d21 * (d11 * ap[j + (k - 1) * ((*n << 1) - k) / 2] - ap[j + k * ((*n << 1) - k - 1) / 2]); wkp1 = d21 * (d22 * ap[j + k * ((*n << 1) - k - 1) / 2] - ap[j + (k - 1) * ((*n << 1) - k) / 2]); i__2 = *n; for (i__ = j; i__ <= i__2; ++i__) { ap[i__ + (j - 1) * ((*n << 1) - j) / 2] = ap[i__ + (j - 1) * ((*n << 1) - j) / 2] - ap[i__ + (k - 1) * ((*n << 1) - k) / 2] * wk - ap[i__ + k * ((*n << 1) - k - 1) / 2] * wkp1; /* L90: */ } ap[j + (k - 1) * ((*n << 1) - k) / 2] = wk; ap[j + k * ((*n << 1) - k - 1) / 2] = wkp1; /* 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 SSPTRF */ } /* ssptrf_ */