.TH "SRC/chesvxx.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/chesvxx.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBchesvxx\fP (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)" .br .RI "\fB CHESVXX computes the solution to system of linear equations A * X = B for HE matrices\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine chesvxx (character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)" .PP \fB CHESVXX computes the solution to system of linear equations A * X = B for HE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> CHESVXX uses the diagonal pivoting factorization to compute the !> solution to a complex system of linear equations A * X = B, where !> A is an N-by-N Hermitian matrix and X and B are N-by-NRHS !> matrices\&. !> !> If requested, both normwise and maximum componentwise error bounds !> are returned\&. CHESVXX will return a solution with a tiny !> guaranteed error (O(eps) where eps is the working machine !> precision) unless the matrix is very ill-conditioned, in which !> case a warning is returned\&. Relevant condition numbers also are !> calculated and returned\&. !> !> CHESVXX accepts user-provided factorizations and equilibration !> factors; see the definitions of the FACT and EQUED options\&. !> Solving with refinement and using a factorization from a previous !> CHESVXX call will also produce a solution with either O(eps) !> errors or warnings, but we cannot make that claim for general !> user-provided factorizations and equilibration factors if they !> differ from what CHESVXX would itself produce\&. !> .fi .PP .RE .PP \fBDescription:\fP .RS 4 .PP .nf !> !> 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 LU decomposition is used to factor !> the matrix A (after equilibration if FACT = 'E') 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 Hermitian and block diagonal with !> 1-by-1 and 2-by-2 diagonal blocks\&. !> !> 3\&. 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 (see !> argument RCOND)\&. If the reciprocal of the condition number is !> less than machine precision, 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\&. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), !> the routine will use iterative refinement to try to get a small !> error and error bounds\&. Refinement calculates the residual to at !> least twice the working precision\&. !> !> 6\&. If equilibration was used, the matrix X is premultiplied by !> diag(R) so that it solves the original system before !> equilibration\&. !> .fi .PP .PP .nf !> Some optional parameters are bundled in the PARAMS array\&. These !> settings determine how refinement is performed, but often the !> defaults are acceptable\&. If the defaults are acceptable, users !> can pass NPARAMS = 0 which prevents the source code from accessing !> the PARAMS argument\&. !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIFACT\fP .PP .nf !> 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, AF and IPIV contain the factored form of A\&. !> If EQUED is not 'N', the matrix A has been !> equilibrated with scaling factors given by S\&. !> A, AF, and IPIV are not modified\&. !> = 'N': The matrix A will be copied to AF and factored\&. !> = 'E': The matrix A will be equilibrated if necessary, then !> copied to AF and factored\&. !> .fi .PP .br \fIUPLO\fP .PP .nf !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The number of linear equations, i\&.e\&., the order of the !> matrix A\&. N >= 0\&. !> .fi .PP .br \fINRHS\fP .PP .nf !> NRHS is INTEGER !> The number of right hand sides, i\&.e\&., the number of columns !> of the matrices B and X\&. NRHS >= 0\&. !> .fi .PP .br \fIA\fP .PP .nf !> A is COMPLEX array, dimension (LDA,N) !> The Hermitian matrix A\&. If UPLO = 'U', the leading N-by-N !> upper triangular part of A contains the upper triangular !> part of the matrix A, and the strictly lower triangular !> part of A is not referenced\&. If UPLO = 'L', the leading !> N-by-N lower triangular part of A contains the lower !> triangular part of the matrix A, and the strictly upper !> triangular part of A is not referenced\&. !> !> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by !> diag(S)*A*diag(S)\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of the array A\&. LDA >= max(1,N)\&. !> .fi .PP .br \fIAF\fP .PP .nf !> AF is COMPLEX array, dimension (LDAF,N) !> If FACT = 'F', then AF 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**H or A = L*D*L**H as computed by CHETRF\&. !> !> If FACT = 'N', then AF is an output argument and on exit !> returns the block diagonal matrix D and the multipliers !> used to obtain the factor U or L from the factorization A = !> U*D*U**H or A = L*D*L**H\&. !> .fi .PP .br \fILDAF\fP .PP .nf !> LDAF is INTEGER !> The leading dimension of the array AF\&. LDAF >= max(1,N)\&. !> .fi .PP .br \fIIPIV\fP .PP .nf !> 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 CHETRF\&. 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 CHETRF\&. !> .fi .PP .br \fIEQUED\fP .PP .nf !> EQUED is CHARACTER*1 !> Specifies the form of equilibration that was done\&. !> = 'N': No equilibration (always true if FACT = 'N')\&. !> = 'Y': Both row and column equilibration, 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\&. !> .fi .PP .br \fIS\fP .PP .nf !> S is REAL array, dimension (N) !> The scale factors for A\&. If EQUED = 'Y', A is multiplied on !> the left and right by diag(S)\&. 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\&. If S is output, each !> element of S is a power of the radix\&. If S is input, each element !> of S should 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\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is COMPLEX array, dimension (LDB,NRHS) !> On entry, the N-by-NRHS right hand side matrix B\&. !> On exit, !> if EQUED = 'N', B is not modified; !> if EQUED = 'Y', B is overwritten by diag(S)*B; !> .fi .PP .br \fILDB\fP .PP .nf !> LDB is INTEGER !> The leading dimension of the array B\&. LDB >= max(1,N)\&. !> .fi .PP .br \fIX\fP .PP .nf !> X is COMPLEX array, dimension (LDX,NRHS) !> If INFO = 0, the N-by-NRHS solution matrix X to the original !> system of equations\&. Note that A and B are modified on exit if !> EQUED \&.ne\&. 'N', and the solution to the equilibrated system is !> inv(diag(S))*X\&. !> .fi .PP .br \fILDX\fP .PP .nf !> LDX is INTEGER !> The leading dimension of the array X\&. LDX >= max(1,N)\&. !> .fi .PP .br \fIRCOND\fP .PP .nf !> RCOND is REAL !> Reciprocal scaled condition number\&. This is an estimate of the !> reciprocal Skeel condition number of the matrix A after !> equilibration (if done)\&. If this is less than the machine !> precision (in particular, if it is zero), the matrix is singular !> to working precision\&. Note that the error may still be small even !> if this number is very small and the matrix appears ill- !> conditioned\&. !> .fi .PP .br \fIRPVGRW\fP .PP .nf !> RPVGRW is REAL !> Reciprocal pivot growth\&. On exit, this contains the reciprocal !> pivot growth factor norm(A)/norm(U)\&. The !> norm is used\&. If this is much less than 1, then the stability of !> the LU factorization of the (equilibrated) matrix A could be poor\&. !> This also means that the solution X, estimated condition numbers, !> and error bounds could be unreliable\&. If factorization fails with !> 0 for the leading INFO columns of A\&. !> .fi .PP .br \fIBERR\fP .PP .nf !> BERR is REAL array, dimension (NRHS) !> Componentwise relative backward error\&. This is 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)\&. !> .fi .PP .br \fIN_ERR_BNDS\fP .PP .nf !> N_ERR_BNDS is INTEGER !> Number of error bounds to return for each right hand side !> and each type (normwise or componentwise)\&. See ERR_BNDS_NORM and !> ERR_BNDS_COMP below\&. !> .fi .PP .br \fIERR_BNDS_NORM\fP .PP .nf !> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) !> For each right-hand side, this array contains information about !> various error bounds and condition numbers corresponding to the !> normwise relative error, which is defined as follows: !> !> Normwise relative error in the ith solution vector: !> max_j (abs(XTRUE(j,i) - X(j,i))) !> ------------------------------ !> max_j abs(X(j,i)) !> !> The array is indexed by the type of error information as described !> below\&. There currently are up to three pieces of information !> returned\&. !> !> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith !> right-hand side\&. !> !> The second index in ERR_BNDS_NORM(:,err) contains the following !> three fields: !> err = 1 boolean\&. Trust the answer if the !> reciprocal condition number is less than the threshold !> sqrt(n) * slamch('Epsilon')\&. !> !> err = 2 error bound: The estimated forward error, !> almost certainly within a factor of 10 of the true error !> so long as the next entry is greater than the threshold !> sqrt(n) * slamch('Epsilon')\&. This error bound should only !> be trusted if the previous boolean is true\&. !> !> err = 3 Reciprocal condition number: Estimated normwise !> reciprocal condition number\&. Compared with the threshold !> sqrt(n) * slamch('Epsilon') to determine if the error !> estimate is \&. These reciprocal condition !> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some !> appropriately scaled matrix Z\&. !> Let Z = S*A, where S scales each row by a power of the !> radix so all absolute row sums of Z are approximately 1\&. !> !> See Lapack Working Note 165 for further details and extra !> cautions\&. !> .fi .PP .br \fIERR_BNDS_COMP\fP .PP .nf !> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) !> For each right-hand side, this array contains information about !> various error bounds and condition numbers corresponding to the !> componentwise relative error, which is defined as follows: !> !> Componentwise relative error in the ith solution vector: !> abs(XTRUE(j,i) - X(j,i)) !> max_j ---------------------- !> abs(X(j,i)) !> !> The array is indexed by the right-hand side i (on which the !> componentwise relative error depends), and the type of error !> information as described below\&. There currently are up to three !> pieces of information returned for each right-hand side\&. If !> componentwise accuracy is not requested (PARAMS(3) = 0\&.0), then !> ERR_BNDS_COMP is not accessed\&. If N_ERR_BNDS < 3, then at most !> the first (:,N_ERR_BNDS) entries are returned\&. !> !> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith !> right-hand side\&. !> !> The second index in ERR_BNDS_COMP(:,err) contains the following !> three fields: !> err = 1 boolean\&. Trust the answer if the !> reciprocal condition number is less than the threshold !> sqrt(n) * slamch('Epsilon')\&. !> !> err = 2 error bound: The estimated forward error, !> almost certainly within a factor of 10 of the true error !> so long as the next entry is greater than the threshold !> sqrt(n) * slamch('Epsilon')\&. This error bound should only !> be trusted if the previous boolean is true\&. !> !> err = 3 Reciprocal condition number: Estimated componentwise !> reciprocal condition number\&. Compared with the threshold !> sqrt(n) * slamch('Epsilon') to determine if the error !> estimate is \&. These reciprocal condition !> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some !> appropriately scaled matrix Z\&. !> Let Z = S*(A*diag(x)), where x is the solution for the !> current right-hand side and S scales each row of !> A*diag(x) by a power of the radix so all absolute row !> sums of Z are approximately 1\&. !> !> See Lapack Working Note 165 for further details and extra !> cautions\&. !> .fi .PP .br \fINPARAMS\fP .PP .nf !> NPARAMS is INTEGER !> Specifies the number of parameters set in PARAMS\&. If <= 0, the !> PARAMS array is never referenced and default values are used\&. !> .fi .PP .br \fIPARAMS\fP .PP .nf !> PARAMS is REAL array, dimension NPARAMS !> Specifies algorithm parameters\&. If an entry is < 0\&.0, then !> that entry will be filled with default value used for that !> parameter\&. Only positions up to NPARAMS are accessed; defaults !> are used for higher-numbered parameters\&. !> !> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative !> refinement or not\&. !> Default: 1\&.0 !> = 0\&.0: No refinement is performed, and no error bounds are !> computed\&. !> = 1\&.0: Use the double-precision refinement algorithm, !> possibly with doubled-single computations if the !> compilation environment does not support DOUBLE !> PRECISION\&. !> (other values are reserved for future use) !> !> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual !> computations allowed for refinement\&. !> Default: 10 !> Aggressive: Set to 100 to permit convergence using approximate !> factorizations or factorizations other than LU\&. If !> the factorization uses a technique other than !> Gaussian elimination, the guarantees in !> err_bnds_norm and err_bnds_comp may no longer be !> trustworthy\&. !> !> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code !> will attempt to find a solution with small componentwise !> relative error in the double-precision algorithm\&. Positive !> is true, 0\&.0 is false\&. !> Default: 1\&.0 (attempt componentwise convergence) !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is COMPLEX array, dimension (5*N) !> .fi .PP .br \fIRWORK\fP .PP .nf !> RWORK is REAL array, dimension (2*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is INTEGER !> = 0: Successful exit\&. The solution to every right-hand side is !> guaranteed\&. !> < 0: If INFO = -i, the i-th argument had an illegal value !> > 0 and <= N: U(INFO,INFO) is exactly zero\&. The factorization !> has been completed, but the factor U is exactly singular, so !> the solution and error bounds could not be computed\&. RCOND = 0 !> is returned\&. !> = N+J: The solution corresponding to the Jth right-hand side is !> not guaranteed\&. The solutions corresponding to other right- !> hand sides K with K > J may not be guaranteed as well, but !> only the first such right-hand side is reported\&. If a small !> componentwise error is not requested (PARAMS(3) = 0\&.0) then !> the Jth right-hand side is the first with a normwise error !> bound that is not guaranteed (the smallest J such !> that ERR_BNDS_NORM(J,1) = 0\&.0)\&. By default (PARAMS(3) = 1\&.0) !> the Jth right-hand side is the first with either a normwise or !> componentwise error bound that is not guaranteed (the smallest !> J such that either ERR_BNDS_NORM(J,1) = 0\&.0 or !> ERR_BNDS_COMP(J,1) = 0\&.0)\&. See the definition of !> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1)\&. To get information !> about all of the right-hand sides check ERR_BNDS_NORM or !> ERR_BNDS_COMP\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB505\fP of file \fBchesvxx\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.