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NAME DSPSVX - use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N- by-NRHS matrices SYNOPSIS SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) CHARACTER FACT, UPLO INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) PURPOSE DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also provided. DESCRIPTION The following steps are performed: 1. If FACT = 'N', the diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = 'U', or A = L * D * L**T, if UPLO = 'L', 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. 2. The factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, steps 3 and 4 are skipped. 3. The system of equations is solved for X using the fac- tored form of A. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. ARGUMENTS FACT (input) CHARACTER*1 Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified. = 'N': The matrix A will be copied to AFP and factored. UPLO (input) CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) INTEGER The number of linear equations, i.e., 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 matrices B and X. NRHS >= 0. AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) 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. See below for further details. AFP (input or output) DOUBLE PRECISION array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the fac- tor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as a packed triangular matrix in the same storage format as A. If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as a packed triangular matrix in the same storage format as A. IPIV (input or output) INTEGER array, dimension (N) If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by DSPTRF. 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 inter- changed 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 inter- changed and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by DSPTRF. B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0, and the solution and error bounds are not computed. FERR (output) DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accu- rate, the error bound is guaranteed. 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 entry of A or B that makes X(j) an exact solution). WORK (workspace) DOUBLE PRECISION array, dimension (3*N) IWORK (workspace) INTEGER array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0 and <= N: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solu- tion and error bounds could not be computed. = N+1: the block diagonal matrix D is nonsingular, but RCOND is less than machine precision. The factori- zation has been completed, but the matrix is singu- lar to working precision, so the solution and error bounds have not been computed. FURTHER DETAILS The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U': Two-dimensional storage of the symmetric matrix A: a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = aji) a44 Packed storage of the upper triangle of A: AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]