Uses extra precise iterative refinement to compute the solution to the system of linear equations with a symmetric or Hermitian positive-definite coefficient matrix A applying the Cholesky factorization.
The routine uses the Cholesky factorization
(real flavors) /
(complex flavors) or
(real flavors) /
(complex flavors) to compute the solution to a real or complex system of linear equations
nreal symmetric/Hermitian positive definite matrix, the columns of matrix
Bare individual right-hand sides, and the columns of
Xare the corresponding solutions.
Both normwise and maximum componentwise error bounds are also provided on request. The routine returns a solution with a small guaranteed error (
epsis the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also calculated and returned.
The routine accepts user-provided factorizations and equilibration factors; see definitions of the
equedoptions. Solving with refinement and using a factorization from a previous call of the routine also produces a solution with
O(eps)errors or warnings but that may not be true for general user-provided factorizations and equilibration factors if they differ from what the routine would itself produce.
?posvxxperforms the following steps:
- If, scaling factors are computed to equilibrate the system:fact='E'diag(s)*A*diag(s) *inv(diag(s))*X=diag(s)*BWhether or not the system will be equilibrated depends on the scaling of the matrixA, but if equilibration is used,Ais overwritten byanddiag(s)*A*diag(s)Bby.diag(s)*B
- Iforfact='N', the Cholesky decomposition is used to factor the matrix'E'A(after equilibration if) asfact='E'(real),A=UT*U(complex), ifA=UH*U,uplo='U'or(real),A=L*LT(complex), ifA=L*LH,uplo='L'whereUis an upper triangular matrix andLis a lower triangular matrix.
- If the leadingi-by-iprincipal minor is not positive-definite, the routine returns with. Otherwise, the factored form ofinfo=iAis used to estimate the condition number of the matrixA(see thercondparameter). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve forXand compute error bounds.
- The system of equations is solved forXusing the factored form ofA.
- By default, unlessparamsis set to zero, the routine applies iterative refinement to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision.
- If equilibration was used, the matrixXis premultiplied byso that it solves the original system before equilibration.diag(s)
- Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
- Must be'F','N', or'E'.Specifies whether or not the factored form of the matrixAis supplied on entry, and if not, whether the matrixAshould be equilibrated before it is factored.If, on entry,fact='F'afcontains the factored form ofA. Ifequedis not'N', the matrixAhas been equilibrated with scaling factors given bys. Parametersaandafare not modified.If, the matrixfact='N'Awill be copied toafand factored.If, the matrixfact='E'Awill be equilibrated, if necessary, copied toafand factored.
- Must be'U'or'L'.Indicates whether the upper or lower triangular part ofAis stored:If, the upper triangle ofuplo='U'Ais stored.If, the lower triangle ofuplo='L'Ais stored.
- The number of linear equations; the order of the matrixA;n≥0.
- The number of right-hand sides; the number of columns of the matricesBandX;nrhs≥0.
- Arrays:a(size max(,lda*n))af(size max(,ldaf*n))b)size max(1,.ldb*nrhs) for column major layout and max(1,ldb*n) for row major layout)The arrayacontains the matrixAas specified byuplo. Ifandfact='F'equed='Y', thenAmust have been equilibrated by the scaling factors ins, andamust contain the equilibrated matrix.diag(s)*A*diag(s)The arrayafis an input argument if. It contains the triangular factorfact='F'UorLfrom the Cholesky factorization ofAin the same storage format asA. Ifequedis not'N', thenafis the factored form of the equilibrated matrix.diag(s)*A*diag(s)The arraybcontains the matrixBwhose columns are the right-hand sides for the systems of equations.
- The leading dimension of the arraya;.lda≥max(1,n)
- The leading dimension of the arrayaf;.ldaf≥max(1,n)
- Must be'N'or'Y'.equedis an input argument if. It specifies the form of equilibration that was done:fact='F'If, no equilibration was done (always true ifequed='N').fact='N'if, both row and column equilibration was done, that is,equed='Y'Ahas been replaced by.diag(s)*A*diag(s)
- Array, size (n). The arrayscontains the scale factors forA. This array is an input argument ifonly; otherwise it is an output argument.fact='F'If,equed='N'sis not accessed.Ifandfact='F'equed='Y', each element ofsmust be positive.Each element ofsshould be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
- The leading dimension of the arrayb;.ldb≥max(1,n) for column major layout andldb≥nrhsfor row major layout
- The leading dimension of the output arrayx;.ldx≥max(1,n) for column major layout andldx≥nrhsfor row major layout
- Number of error bounds to return for each right hand side and each type (normwise or componentwise). Seeerr_bnds_normanderr_bnds_compdescriptions in theOutput Argumentssection below.
- Specifies the number of parameters set inparams. If≤0, theparamsarray is never referenced and default values are used.
- Array, sizemax(1,. Specifies algorithm parameters. If an entry is less than 0.0, that entry is filled with the default value used for that parameter. Only positions up tonparams)nparamsare accessed; defaults are used for higher-numbered parameters. If defaults are acceptable, you can passnparams= 0, which prevents the source code from accessing theparamsargument.: Whether to perform iterative refinement or not. Default: 1.0 (for single precision flavors), 1.0D+0 (for double precision flavors).params
(Other values are reserved for future use.): Maximum number of residual computations allowed for refinement.params
- No refinement is performed and no error bounds are computed.
- Use the extra-precise refinement algorithm.
: Flag determining if the code will attempt to find a solution with a small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence).params
- Set to 100.0 to permit convergence using approximate factorizations or factorizations other thanLU. If the factorization uses a technique other than Gaussian elimination, the guarantees inerr_bnds_normanderr_bnds_compmay no longer be trustworthy.
- Array, sizemax(1,.ldx*nrhs) for column major layout and max(1,ldx*n) for row major layoutIf, the arrayinfo= 0xcontains the solutionn-by-nrhsmatrixXto the original system of equations. Note thatAandBare modified on exit if, and the solution to the equilibrated system is:equed≠'N'inv(.diag(s))*X
- Arrayais not modified on exit iforfact='F''N', or ifandfact='E'equed='N'.Ifandfact='E'equed='Y',Ais overwritten by.diag(s)*A*diag(s)
- Iforfact='N''E', thenafis an output argument and on exit returns the triangular factorUorLfrom the Cholesky factorizationorA=UT*U(real routines),A=L*LTorA=UH*U(complex routines) of the original matrixA=L*LHA(if), or of the equilibrated matrixfact='N'