.TH CSPSV 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME CSPSV - the solution to a complex system of linear equations A * X = B, .SH SYNOPSIS .TP 18 SUBROUTINE CSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO ) .TP 18 .ti +4 CHARACTER UPLO .TP 18 .ti +4 INTEGER INFO, LDB, N, NRHS .TP 18 .ti +4 INTEGER IPIV( * ) .TP 18 .ti +4 COMPLEX AP( * ), B( LDB, * ) .SH PURPOSE CSPSV computes the solution to a complex 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. .br The diagonal pivoting method is used to factor A as .br A = U * D * U**T, if UPLO = \(aqU\(aq, or .br A = L * D * L**T, if UPLO = \(aqL\(aq, .br where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B. .br .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 = \(aqU\(aq: Upper triangle of A is stored; .br = \(aqL\(aq: Lower triangle of A is stored. .TP 8 N (input) INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. .TP 8 NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. .TP 8 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 = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = \(aqL\(aq, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. See below for further details. On exit, 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. .TP 8 IPIV (output) INTEGER array, dimension (N) 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 = \(aqU\(aq 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 = \(aqL\(aq 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. .TP 8 B (input/output) COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: 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 solution could not be computed. .SH FURTHER DETAILS The packed storage scheme is illustrated by the following example when N = 4, UPLO = \(aqU\(aq: .br Two-dimensional storage of the symmetric matrix A: .br a11 a12 a13 a14 .br a22 a23 a24 .br a33 a34 (aij = aji) .br a44 .br Packed storage of the upper triangle of A: .br AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]