LAPACK 3.12.0
LAPACK: Linear Algebra PACKage

subroutine sppsvx  (  character  fact, 
character  uplo,  
integer  n,  
integer  nrhs,  
real, dimension( * )  ap,  
real, dimension( * )  afp,  
character  equed,  
real, dimension( * )  s,  
real, dimension( ldb, * )  b,  
integer  ldb,  
real, dimension( ldx, * )  x,  
integer  ldx,  
real  rcond,  
real, dimension( * )  ferr,  
real, dimension( * )  berr,  
real, dimension( * )  work,  
integer, dimension( * )  iwork,  
integer  info  
) 
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
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SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an NbyN symmetric positive definite matrix stored in packed format and X and B are NbyNRHS matrices. Error bounds on the solution and a condition estimate are also provided.
The following steps are performed: 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. 3. If the leading principal minor of order i is not positive, 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. 4. The system of equations is solved for X using the factored form of A. 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.
[in]  FACT  FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified. = 'N': The matrix A will be copied to AFP and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AFP and factored. 
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 
[in,out]  AP  AP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. See below for further details. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S). 
[in,out]  AFP  AFP is REAL array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A. If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the original matrix A. If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix). 
[in,out]  EQUED  EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. 
[in,out]  S  S is REAL array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the NbyNRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  X  X is REAL array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the NbyNRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  RCOND  RCOND is REAL The estimate of the reciprocal condition number of the matrix A after equilibration (if done). 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. 
[out]  FERR  FERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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. 
[out]  BERR  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). 
[out]  WORK  WORK is REAL array, dimension (3*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, and i is <= N: the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U 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. 
The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U': Twodimensional storage of the symmetric matrix A: a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = conjg(aji)) a44 Packed storage of the upper triangle of A: AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
Definition at line 309 of file sppsvx.f.