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NAME CSPSVX - use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a com- plex 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 CSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO ) CHARACTER FACT, UPLO INTEGER INFO, LDB, LDX, N, NRHS REAL RCOND INTEGER IPIV( * ) REAL BERR( * ), FERR( * ), RWORK( * ) COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * ) PURPOSE CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N sym- metric 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) COMPLEX 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) COMPLEX 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 factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, 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 CSPTRF, 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 CSPTRF. 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 CSPTRF. B (input) COMPLEX 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) COMPLEX 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) REAL 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) REAL 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) REAL 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) COMPLEX array, dimension (2*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 > 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 ]