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template<typename T>
void spsvx(char *fact, char *uplo, integer *n, integer *nrhs, T *ap, T *afp, integer *ipiv, T *b, integer *ldb, T *x, integer *ldx, T *rcond, T *ferr, T *berr, T *work, integer *iwork, integer *info)# SPSVX computes the solution to system of linear equations A * X = B for OTHER matrices.
Purpose:
SPSVX 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. * \b 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. 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. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
- Parameters:
FACT – [in]
FACT is 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 – [in]
UPLO is CHARACTER*1
= ‘U’: Upper triangle of A is stored;
= ‘L’: Lower triangle of A is stored.N – [in]
N is INTEGER
The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS – [in]
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AP – [in]
AP is 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 = ‘U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ‘L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.AFP – [inout]
AFP is REAL 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 SSPTRF, 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 SSPTRF, stored as a packed triangular matrix in the same storage format as A.IPIV – [inout]
IPIV is 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 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 = ‘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.
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 SSPTRF.B – [in]
B is REAL array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB – [in]
LDB is INTEGER
The leading dimension of the array B. LDB >= fla_max(1,N).
X – [out]
X is REAL array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX – [in]
LDX is INTEGER
The leading dimension of the array X. LDX >= fla_max(1,N).
RCOND – [out]
RCOND is 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.
FERR – [out]
FERR is 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.
BERR – [out]
BERR is 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).
WORK – [out] WORK is REAL array, dimension (3*N)
IWORK – [out] IWORK is INTEGER array, dimension (N)
INFO – [out]
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= 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.