- Generated by
1.15.0
Functions | |
| subroutine | sgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) |
| SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) | |
| subroutine | sgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) |
| SGBSVX computes the solution to system of linear equations A * X = B for GB matrices | |
| subroutine | sgbsvxx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info) |
| SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices | |
This is the group of real solve driver functions for GB matrices
| subroutine sgbsv | ( | integer | n, |
| integer | kl, | ||
| integer | ku, | ||
| integer | nrhs, | ||
| real, dimension( ldab, * ) | ab, | ||
| integer | ldab, | ||
| integer, dimension( * ) | ipiv, | ||
| real, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| integer | info ) |
SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
Download SGBSV + dependencies [TGZ] [ZIP] [TXT]
!> !> SGBSV computes the solution to a real system of linear equations !> A * X = B, where A is a band matrix of order N with KL subdiagonals !> and KU superdiagonals, and X and B are N-by-NRHS matrices. !> !> The LU decomposition with partial pivoting and row interchanges is !> used to factor A as A = L * U, where L is a product of permutation !> and unit lower triangular matrices with KL subdiagonals, and U is !> upper triangular with KL+KU superdiagonals. The factored form of A !> is then used to solve the system of equations A * X = B. !>
| [in] | N | !> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !> |
| [in] | KL | !> KL is INTEGER !> The number of subdiagonals within the band of A. KL >= 0. !> |
| [in] | KU | !> KU is INTEGER !> The number of superdiagonals within the band of A. KU >= 0. !> |
| [in] | NRHS | !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !> |
| [in,out] | AB | !> AB is REAL array, dimension (LDAB,N) !> On entry, the matrix A in band storage, in rows KL+1 to !> 2*KL+KU+1; rows 1 to KL of the array need not be set. !> The j-th column of A is stored in the j-th column of the !> array AB as follows: !> AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL) !> On exit, details of the factorization: U is stored as an !> upper triangular band matrix with KL+KU superdiagonals in !> rows 1 to KL+KU+1, and the multipliers used during the !> factorization are stored in rows KL+KU+2 to 2*KL+KU+1. !> See below for further details. !> |
| [in] | LDAB | !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= 2*KL+KU+1. !> |
| [out] | IPIV | !> IPIV is INTEGER array, dimension (N) !> The pivot indices that define the permutation matrix P; !> row i of the matrix was interchanged with row IPIV(i). !> |
| [in,out] | B | !> B is REAL array, dimension (LDB,NRHS) !> On entry, the N-by-NRHS right hand side matrix B. !> On exit, if INFO = 0, the N-by-NRHS solution matrix X. !> |
| [in] | LDB | !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> |
| [out] | INFO | !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, U(i,i) is exactly zero. The factorization !> has been completed, but the factor U is exactly !> singular, and the solution has not been computed. !> |
!> !> The band storage scheme is illustrated by the following example, when !> M = N = 6, KL = 2, KU = 1: !> !> On entry: On exit: !> !> * * * + + + * * * u14 u25 u36 !> * * + + + + * * u13 u24 u35 u46 !> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 !> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 !> a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * !> a31 a42 a53 a64 * * m31 m42 m53 m64 * * !> !> Array elements marked * are not used by the routine; elements marked !> + need not be set on entry, but are required by the routine to store !> elements of U because of fill-in resulting from the row interchanges. !>
Definition at line 161 of file sgbsv.f.
| subroutine sgbsvx | ( | character | fact, |
| character | trans, | ||
| integer | n, | ||
| integer | kl, | ||
| integer | ku, | ||
| integer | nrhs, | ||
| real, dimension( ldab, * ) | ab, | ||
| integer | ldab, | ||
| real, dimension( ldafb, * ) | afb, | ||
| integer | ldafb, | ||
| integer, dimension( * ) | ipiv, | ||
| character | equed, | ||
| real, dimension( * ) | r, | ||
| real, dimension( * ) | c, | ||
| 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 ) |
SGBSVX computes the solution to system of linear equations A * X = B for GB matrices
Download SGBSVX + dependencies [TGZ] [ZIP] [TXT]
!> !> SGBSVX uses the LU factorization to compute the solution to a real !> system of linear equations A * X = B, A**T * X = B, or A**H * X = B, !> where A is a band matrix of order N with KL subdiagonals and KU !> superdiagonals, and X and B are N-by-NRHS matrices. !> !> Error bounds on the solution and a condition estimate are also !> provided. !>
!> !> The following steps are performed by this subroutine: !> !> 1. If FACT = 'E', real scaling factors are computed to equilibrate !> the system: !> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B !> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B !> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') !> or diag(C)*B (if TRANS = 'T' or 'C'). !> !> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the !> matrix A (after equilibration if FACT = 'E') as !> A = L * U, !> where L is a product of permutation and unit lower triangular !> matrices with KL subdiagonals, and U is upper triangular with !> KL+KU superdiagonals. !> !> 3. If some U(i,i)=0, so that U 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. !> !> 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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, AFB and IPIV contain the factored form of !> A. If EQUED is not 'N', the matrix A has been !> equilibrated with scaling factors given by R and C. !> AB, AFB, and IPIV are not modified. !> = 'N': The matrix A will be copied to AFB and factored. !> = 'E': The matrix A will be equilibrated if necessary, then !> copied to AFB and factored. !> |
| [in] | TRANS | !> TRANS is CHARACTER*1 !> Specifies the form of the system of equations. !> = 'N': A * X = B (No transpose) !> = 'T': A**T * X = B (Transpose) !> = 'C': A**H * X = B (Transpose) !> |
| [in] | N | !> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !> |
| [in] | KL | !> KL is INTEGER !> The number of subdiagonals within the band of A. KL >= 0. !> |
| [in] | KU | !> KU is INTEGER !> The number of superdiagonals within the band of A. KU >= 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] | AB | !> AB is REAL array, dimension (LDAB,N) !> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. !> The j-th column of A is stored in the j-th column of the !> array AB as follows: !> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) !> !> If FACT = 'F' and EQUED is not 'N', then A must have been !> equilibrated by the scaling factors in R and/or C. AB is not !> modified if FACT = 'F' or 'N', or if FACT = 'E' and !> EQUED = 'N' on exit. !> !> On exit, if EQUED .ne. 'N', A is scaled as follows: !> EQUED = 'R': A := diag(R) * A !> EQUED = 'C': A := A * diag(C) !> EQUED = 'B': A := diag(R) * A * diag(C). !> |
| [in] | LDAB | !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KL+KU+1. !> |
| [in,out] | AFB | !> AFB is REAL array, dimension (LDAFB,N) !> If FACT = 'F', then AFB is an input argument and on entry !> contains details of the LU factorization of the band matrix !> A, as computed by SGBTRF. U is stored as an upper triangular !> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, !> and the multipliers used during the factorization are stored !> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is !> the factored form of the equilibrated matrix A. !> !> If FACT = 'N', then AFB is an output argument and on exit !> returns details of the LU factorization of A. !> !> If FACT = 'E', then AFB is an output argument and on exit !> returns details of the LU factorization of the equilibrated !> matrix A (see the description of AB for the form of the !> equilibrated matrix). !> |
| [in] | LDAFB | !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. !> |
| [in,out] | IPIV | !> IPIV is INTEGER array, dimension (N) !> If FACT = 'F', then IPIV is an input argument and on entry !> contains the pivot indices from the factorization A = L*U !> as computed by SGBTRF; row i of the matrix was interchanged !> with row IPIV(i). !> !> If FACT = 'N', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = L*U !> of the original matrix A. !> !> If FACT = 'E', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = L*U !> of the equilibrated matrix A. !> |
| [in,out] | EQUED | !> EQUED is CHARACTER*1 !> Specifies the form of equilibration that was done. !> = 'N': No equilibration (always true if FACT = 'N'). !> = 'R': Row equilibration, i.e., A has been premultiplied by !> diag(R). !> = 'C': Column equilibration, i.e., A has been postmultiplied !> by diag(C). !> = 'B': Both row and column equilibration, i.e., A has been !> replaced by diag(R) * A * diag(C). !> EQUED is an input argument if FACT = 'F'; otherwise, it is an !> output argument. !> |
| [in,out] | R | !> R is REAL array, dimension (N) !> The row scale factors for A. If EQUED = 'R' or 'B', A is !> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R !> is not accessed. R is an input argument if FACT = 'F'; !> otherwise, R is an output argument. If FACT = 'F' and !> EQUED = 'R' or 'B', each element of R must be positive. !> |
| [in,out] | C | !> C is REAL array, dimension (N) !> The column scale factors for A. If EQUED = 'C' or 'B', A is !> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C !> is not accessed. C is an input argument if FACT = 'F'; !> otherwise, C is an output argument. If FACT = 'F' and !> EQUED = 'C' or 'B', each element of C must be positive. !> |
| [in,out] | B | !> B is REAL array, dimension (LDB,NRHS) !> On entry, the right hand side matrix B. !> On exit, !> if EQUED = 'N', B is not modified; !> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by !> diag(R)*B; !> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is !> overwritten by diag(C)*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 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(C))*X if TRANS = 'N' and !> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' !> and EQUED = 'R' or 'B'. !> |
| [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 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. !> |
| [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) !> On exit, WORK(1) contains the reciprocal pivot growth !> factor norm(A)/norm(U). The norm is !> used. If WORK(1) 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, condition !> estimator RCOND, and forward error bound FERR could be !> unreliable. If factorization fails with 0<INFO<=N, then !> WORK(1) contains the reciprocal pivot growth factor for the !> leading INFO columns of A. !> |
| [out] | IWORK | !> IWORK is INTEGER array, dimension (N) !> |
| [out] | INFO | !> 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: U(i,i) 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+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 !> |
Definition at line 365 of file sgbsvx.f.
| subroutine sgbsvxx | ( | character | fact, |
| character | trans, | ||
| integer | n, | ||
| integer | kl, | ||
| integer | ku, | ||
| integer | nrhs, | ||
| real, dimension( ldab, * ) | ab, | ||
| integer | ldab, | ||
| real, dimension( ldafb, * ) | afb, | ||
| integer | ldafb, | ||
| integer, dimension( * ) | ipiv, | ||
| character | equed, | ||
| real, dimension( * ) | r, | ||
| real, dimension( * ) | c, | ||
| real, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| real, 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, | ||
| real, dimension( * ) | work, | ||
| integer, dimension( * ) | iwork, | ||
| integer | info ) |
SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices
Download SGBSVXX + dependencies [TGZ] [ZIP] [TXT]
!> !> SGBSVXX uses the LU factorization to compute the solution to a !> real system of linear equations A * X = B, where A is an !> N-by-N matrix and X and B are N-by-NRHS matrices. !> !> If requested, both normwise and maximum componentwise error bounds !> are returned. SGBSVXX 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. !> !> SGBSVXX 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 !> SGBSVXX 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 SGBSVXX would itself produce. !>
!> !> The following steps are performed: !> !> 1. If FACT = 'E', real scaling factors are computed to equilibrate !> the system: !> !> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B !> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B !> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') !> or diag(C)*B (if TRANS = 'T' or 'C'). !> !> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor !> the matrix A (after equilibration if FACT = 'E') as !> !> A = P * L * U, !> !> where P is a permutation matrix, L is a unit lower triangular !> matrix, and U is upper triangular. !> !> 3. If some U(i,i)=0, so that U 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so !> that it solves the original system before equilibration. !>
!> 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. !>
| [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, 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 R and C. !> 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. !> |
| [in] | TRANS | !> TRANS is CHARACTER*1 !> Specifies the form of the system of equations: !> = 'N': A * X = B (No transpose) !> = 'T': A**T * X = B (Transpose) !> = 'C': A**H * X = B (Conjugate Transpose = Transpose) !> |
| [in] | N | !> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !> |
| [in] | KL | !> KL is INTEGER !> The number of subdiagonals within the band of A. KL >= 0. !> |
| [in] | KU | !> KU is INTEGER !> The number of superdiagonals within the band of A. KU >= 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] | AB | !> AB is REAL array, dimension (LDAB,N) !> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. !> The j-th column of A is stored in the j-th column of the !> array AB as follows: !> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) !> !> If FACT = 'F' and EQUED is not 'N', then AB must have been !> equilibrated by the scaling factors in R and/or C. AB is not !> modified if FACT = 'F' or 'N', or if FACT = 'E' and !> EQUED = 'N' on exit. !> !> On exit, if EQUED .ne. 'N', A is scaled as follows: !> EQUED = 'R': A := diag(R) * A !> EQUED = 'C': A := A * diag(C) !> EQUED = 'B': A := diag(R) * A * diag(C). !> |
| [in] | LDAB | !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KL+KU+1. !> |
| [in,out] | AFB | !> AFB is REAL array, dimension (LDAFB,N) !> If FACT = 'F', then AFB is an input argument and on entry !> contains details of the LU factorization of the band matrix !> A, as computed by SGBTRF. U is stored as an upper triangular !> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, !> and the multipliers used during the factorization are stored !> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is !> the factored form of the equilibrated matrix A. !> !> If FACT = 'N', then AF is an output argument and on exit !> returns the factors L and U from the factorization A = P*L*U !> of the original matrix A. !> !> If FACT = 'E', then AF is an output argument and on exit !> returns the factors L and U from the factorization A = P*L*U !> of the equilibrated matrix A (see the description of A for !> the form of the equilibrated matrix). !> |
| [in] | LDAFB | !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. !> |
| [in,out] | IPIV | !> IPIV is INTEGER array, dimension (N) !> If FACT = 'F', then IPIV is an input argument and on entry !> contains the pivot indices from the factorization A = P*L*U !> as computed by SGETRF; row i of the matrix was interchanged !> with row IPIV(i). !> !> If FACT = 'N', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = P*L*U !> of the original matrix A. !> !> If FACT = 'E', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = P*L*U !> of the equilibrated matrix A. !> |
| [in,out] | EQUED | !> EQUED is CHARACTER*1 !> Specifies the form of equilibration that was done. !> = 'N': No equilibration (always true if FACT = 'N'). !> = 'R': Row equilibration, i.e., A has been premultiplied by !> diag(R). !> = 'C': Column equilibration, i.e., A has been postmultiplied !> by diag(C). !> = 'B': Both row and column equilibration, i.e., A has been !> replaced by diag(R) * A * diag(C). !> EQUED is an input argument if FACT = 'F'; otherwise, it is an !> output argument. !> |
| [in,out] | R | !> R is REAL array, dimension (N) !> The row scale factors for A. If EQUED = 'R' or 'B', A is !> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R !> is not accessed. R is an input argument if FACT = 'F'; !> otherwise, R is an output argument. If FACT = 'F' and !> EQUED = 'R' or 'B', each element of R must be positive. !> If R is output, each element of R is a power of the radix. !> If R is input, each element of R 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. !> |
| [in,out] | C | !> C is REAL array, dimension (N) !> The column scale factors for A. If EQUED = 'C' or 'B', A is !> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C !> is not accessed. C is an input argument if FACT = 'F'; !> otherwise, C is an output argument. If FACT = 'F' and !> EQUED = 'C' or 'B', each element of C must be positive. !> If C is output, each element of C is a power of the radix. !> If C is input, each element of C 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. !> |
| [in,out] | B | !> B is REAL 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by !> diag(R)*B; !> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is !> overwritten by diag(C)*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, 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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or !> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. !> |
| [in] | LDX | !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> |
| [out] | RCOND | !> 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. !> |
| [out] | RPVGRW | !> 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<INFO<=N, then this contains the reciprocal pivot growth factor !> for the leading INFO columns of A. In SGESVX, this quantity is !> returned in WORK(1). !> |
| [out] | BERR | !> 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). !> |
| [in] | N_ERR_BNDS | !> 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. !> |
| [out] | ERR_BNDS_NORM | !> 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.
!> |
| [out] | ERR_BNDS_COMP | !> 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.
!> |
| [in] | NPARAMS | !> NPARAMS is INTEGER !> Specifies the number of parameters set in PARAMS. If <= 0, the !> PARAMS array is never referenced and default values are used. !> |
| [in,out] | PARAMS | !> 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) !> |
| [out] | WORK | !> WORK is REAL array, dimension (4*N) !> |
| [out] | IWORK | !> IWORK is INTEGER array, dimension (N) !> |
| [out] | INFO | !> 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. !> |
Definition at line 558 of file sgbsvxx.f.
1.15.0