*> \brief SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGBSVXX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * 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 ) * * .. Scalar Arguments .. * CHARACTER EQUED, FACT, TRANS * INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, * $ N_ERR_BNDS * REAL RCOND, RPVGRW * .. * .. Array Arguments .. * INTEGER IPIV( * ), IWORK( * ) * REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), * $ X( LDX , * ),WORK( * ) * REAL R( * ), C( * ), PARAMS( * ), BERR( * ), * $ ERR_BNDS_NORM( NRHS, * ), * $ ERR_BNDS_COMP( NRHS, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * *> \par Description: * ================= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \verbatim *> 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. *> \endverbatim *> *> \param[in] FACT *> \verbatim *> 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. *> \endverbatim *> *> \param[in] TRANS *> \verbatim *> 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) *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of linear equations, i.e., the order of the *> matrix A. N >= 0. *> \endverbatim *> *> \param[in] KL *> \verbatim *> KL is INTEGER *> The number of subdiagonals within the band of A. KL >= 0. *> \endverbatim *> *> \param[in] KU *> \verbatim *> KU is INTEGER *> The number of superdiagonals within the band of A. KU >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of columns *> of the matrices B and X. NRHS >= 0. *> \endverbatim *> *> \param[in,out] AB *> \verbatim *> 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). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KL+KU+1. *> \endverbatim *> *> \param[in,out] AFB *> \verbatim *> 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). *> \endverbatim *> *> \param[in] LDAFB *> \verbatim *> LDAFB is INTEGER *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. *> \endverbatim *> *> \param[in,out] IPIV *> \verbatim *> 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. *> \endverbatim *> *> \param[in,out] EQUED *> \verbatim *> 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. *> \endverbatim *> *> \param[in,out] R *> \verbatim *> 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. *> \endverbatim *> *> \param[in,out] C *> \verbatim *> 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. *> \endverbatim *> *> \param[in,out] B *> \verbatim *> 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. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] X *> \verbatim *> 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'. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the array X. LDX >= max(1,N). *> \endverbatim *> *> \param[out] RCOND *> \verbatim *> 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. *> \endverbatim *> *> \param[out] RPVGRW *> \verbatim *> RPVGRW is REAL *> Reciprocal pivot growth. On exit, this contains the reciprocal *> pivot growth factor norm(A)/norm(U). The "max absolute element" *> 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. In SGESVX, this quantity is *> returned in WORK(1). *> \endverbatim *> *> \param[out] BERR *> \verbatim *> 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). *> \endverbatim *> *> \param[in] N_ERR_BNDS *> \verbatim *> 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. *> \endverbatim *> *> \param[out] ERR_BNDS_NORM *> \verbatim *> 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 "Trust/don't trust" boolean. Trust the answer if the *> reciprocal condition number is less than the threshold *> sqrt(n) * slamch('Epsilon'). *> *> err = 2 "Guaranteed" 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 "guaranteed". 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. *> \endverbatim *> *> \param[out] ERR_BNDS_COMP *> \verbatim *> 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 .LT. 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 "Trust/don't trust" boolean. Trust the answer if the *> reciprocal condition number is less than the threshold *> sqrt(n) * slamch('Epsilon'). *> *> err = 2 "Guaranteed" 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 "guaranteed". 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. *> \endverbatim *> *> \param[in] NPARAMS *> \verbatim *> NPARAMS is INTEGER *> Specifies the number of parameters set in PARAMS. If .LE. 0, the *> PARAMS array is never referenced and default values are used. *> \endverbatim *> *> \param[in,out] PARAMS *> \verbatim *> PARAMS is REAL array, dimension NPARAMS *> Specifies algorithm parameters. If an entry is .LT. 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) *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (4*N) *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date April 2012 * *> \ingroup realGBsolve * * ===================================================================== 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 ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * April 2012 * * .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, $ N_ERR_BNDS REAL RCOND, RPVGRW * .. * .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ X( LDX , * ),WORK( * ) REAL R( * ), C( * ), PARAMS( * ), BERR( * ), $ ERR_BNDS_NORM( NRHS, * ), $ ERR_BNDS_COMP( NRHS, * ) * .. * * ================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I INTEGER CMP_ERR_I, PIV_GROWTH_I PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, $ BERR_I = 3 ) PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, $ PIV_GROWTH_I = 9 ) * .. * .. Local Scalars .. LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU INTEGER INFEQU, I, J, KL, KU REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, $ ROWCND, SMLNUM * .. * .. External Functions .. EXTERNAL LSAME, SLAMCH, SLA_GBRPVGRW LOGICAL LSAME REAL SLAMCH, SLA_GBRPVGRW * .. * .. External Subroutines .. EXTERNAL SGBEQUB, SGBTRF, SGBTRS, SLACPY, SLAQGB, $ XERBLA, SLASCL2, SGBRFSX * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) SMLNUM = SLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF * * Default is failure. If an input parameter is wrong or * factorization fails, make everything look horrible. Only the * pivot growth is set here, the rest is initialized in SGBRFSX. * RPVGRW = ZERO * * Test the input parameters. PARAMS is not tested until SGBRFSX. * IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. $ LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. $ LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KL.LT.0 ) THEN INFO = -4 ELSE IF( KU.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -8 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -10 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -12 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 10 J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) 10 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -13 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -14 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -15 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -16 END IF END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGBSVXX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL SGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL SLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF * * If the scaling factors are not applied, set them to 1.0. * IF ( .NOT.ROWEQU ) THEN DO J = 1, N R( J ) = 1.0 END DO END IF IF ( .NOT.COLEQU ) THEN DO J = 1, N C( J ) = 1.0 END DO END IF END IF * * Scale the right hand side. * IF( NOTRAN ) THEN IF( ROWEQU ) CALL SLASCL2(N, NRHS, R, B, LDB) ELSE IF( COLEQU ) CALL SLASCL2(N, NRHS, C, B, LDB) END IF * IF( NOFACT .OR. EQUIL ) THEN * * Compute the LU factorization of A. * DO 40, J = 1, N DO 30, I = KL+1, 2*KL+KU+1 AFB( I, J ) = AB( I-KL, J ) 30 CONTINUE 40 CONTINUE CALL SGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) * * Return if INFO is non-zero. * IF( INFO.GT.0 ) THEN * * Pivot in column INFO is exactly 0 * Compute the reciprocal pivot growth factor of the * leading rank-deficient INFO columns of A. * RPVGRW = SLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB, $ LDAFB ) RETURN END IF END IF * * Compute the reciprocal pivot growth factor RPVGRW. * RPVGRW = SLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB ) * * Compute the solution matrix X. * CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL SGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, $ INFO ) * * Use iterative refinement to improve the computed solution and * compute error bounds and backward error estimates for it. * CALL SGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, $ WORK, IWORK, INFO ) * * Scale solutions. * IF ( COLEQU .AND. NOTRAN ) THEN CALL SLASCL2 ( N, NRHS, C, X, LDX ) ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN CALL SLASCL2 ( N, NRHS, R, X, LDX ) END IF * RETURN * * End of SGBSVXX * END