.TH SSPSVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME SSPSVX - 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 .SH SYNOPSIS .TP 19 SUBROUTINE SSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) .TP 19 .ti +4 CHARACTER FACT, UPLO .TP 19 .ti +4 INTEGER INFO, LDB, LDX, N, NRHS .TP 19 .ti +4 REAL RCOND .TP 19 .ti +4 INTEGER IPIV( * ), IWORK( * ) .TP 19 .ti +4 REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) .SH PURPOSE SSPSVX 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. .br .SH DESCRIPTION The following steps are performed: .br 1. If FACT = \(aqN\(aq, the diagonal pivoting method is used to factor A as 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 and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. .br 2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, 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, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. 3. The system of equations is solved for X using the factored form of A. .br 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. .br .SH ARGUMENTS .TP 8 FACT (input) CHARACTER*1 Specifies whether or not the factored form of A has been supplied on entry. = \(aqF\(aq: On entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified. = \(aqN\(aq: The matrix A will be copied to AFP and factored. .TP 8 UPLO (input) CHARACTER*1 .br = \(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 matrices B and X. NRHS >= 0. .TP 8 AP (input) REAL 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 = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = \(aqL\(aq, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details. .TP 8 AFP (input or output) REAL array, dimension (N*(N+1)/2) If FACT = \(aqF\(aq, 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 SSPTRF, stored as a packed triangular matrix in the same storage format as A. If FACT = \(aqN\(aq, 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 SSPTRF, stored as a packed triangular matrix in the same storage format as A. .TP 8 IPIV (input or output) INTEGER array, dimension (N) If FACT = \(aqF\(aq, then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by SSPTRF. 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. If FACT = \(aqN\(aq, then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by SSPTRF. .TP 8 B (input) REAL array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 X (output) REAL array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. .TP 8 LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). .TP 8 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. .TP 8 FERR (output) REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. .TP 8 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 element of A or B that makes X(j) an exact solution). .TP 8 WORK (workspace) REAL array, dimension (3*N) .TP 8 IWORK (workspace) INTEGER array, dimension (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, and i is .br <= N: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. .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 ]