sgbsvxx (l)  Linux Manuals
sgbsvxx: SGBSVXX use the LU factorization to compute the solution to a real system of linear equations A * X = B, where A is an NbyN matrix and X and B are NbyNRHS matrices
NAME
SGBSVXX  SGBSVXX use the LU factorization to compute the solution to a real system of linear equations A * X = B, where A is an NbyN matrix and X and B are NbyNRHS matricesSYNOPSIS
 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 )
 IMPLICIT NONE
 CHARACTER EQUED, FACT, TRANS
 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS
 REAL RCOND, RPVGRW
 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, * )
PURPOSE
SGBSVXX uses the LU factorization to compute the solution to a
real system of linear equations
NbyN matrix and X and B are NbyNRHS matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. SGBSVXX will return a solution with a tiny
guaranteed error
precision)
case a warning is returned. Relevant condition numbers also are
calculated and returned.
SGBSVXX accepts userprovided factorizations and equilibration
factors;
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
userprovided factorizations and equilibration factors if they
differ from what SGBSVXX would itself produce.
DESCRIPTION
The following steps are performed:
1. If FACT
the system:
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)
or diag(C)*B
2. If FACT
the matrix A
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
is used to estimate the condition number of the matrix A
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
the routine will use iterative refinement to try to get a small
error and error bounds.
least twice the working precision.
6. If equilibration was used, the matrix X is premultiplied by
diag(C)
that it solves the original system before equilibration.
ARGUMENTS
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. FACT (input) 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.
= aqFaq: On entry, AF and IPIV contain the factored form of A.
If EQUED is not aqNaq, the matrix A has been
equilibrated with scaling factors given by R and C.
A, AF, and IPIV are not modified.
= aqNaq: The matrix A will be copied to AF and factored.
= aqEaq: The matrix A will be equilibrated if necessary, then copied to AF and factored.  TRANS (input) CHARACTER*1

Specifies the form of the system of equations:
= aqNaq: A * X = B (No transpose)
= aqTaq: A**T * X = B (Transpose)
= aqCaq: A**H * X = B (Conjugate Transpose = Transpose)  N (input) INTEGER
 The number of linear equations, i.e., the order of the matrix A. N >= 0.
 KL (input) INTEGER
 The number of subdiagonals within the band of A. KL >= 0.
 KU (input) INTEGER
 The number of superdiagonals within the band of A. KU >= 0.
 NRHS (input) INTEGER
 The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
 AB (input/output) REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
The jth column of A is stored in the jth column of the
array AB as follows:
AB(KU+1+ij,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl)
If FACT = aqFaq and EQUED is not aqNaq, then AB must have been
equilibrated by the scaling factors in R and/or C. AB is not
modified if FACT = aqFaq or aqNaq, or if FACT = aqEaq and
EQUED = aqNaq on exit.
On exit, if EQUED .ne. aqNaq, A is scaled as follows:
EQUED = aqRaq: A := diag(R) * A
EQUED = aqCaq: A := A * diag(C)
EQUED = aqBaq: A := diag(R) * A * diag(C).  LDAB (input) INTEGER
 The leading dimension of the array AB. LDAB >= KL+KU+1.
 AFB (input or output) REAL array, dimension (LDAFB,N)
 If FACT = aqFaq, 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. aqNaq, then AFB is the factored form of the equilibrated matrix A. If FACT = aqNaq, 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 = aqEaq, 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).
 LDAFB (input) INTEGER
 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
 IPIV (input or output) INTEGER array, dimension (N)
 If FACT = aqFaq, 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 = aqNaq, 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 = aqEaq, 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.
 EQUED (input or output) CHARACTER*1

Specifies the form of equilibration that was done.
= aqNaq: No equilibration (always true if FACT = aqNaq).
= aqRaq: Row equilibration, i.e., A has been premultiplied by diag(R). = aqCaq: Column equilibration, i.e., A has been postmultiplied by diag(C). = aqBaq: Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C). EQUED is an input argument if FACT = aqFaq; otherwise, it is an output argument.  R (input or output) REAL array, dimension (N)
 The row scale factors for A. If EQUED = aqRaq or aqBaq, A is multiplied on the left by diag(R); if EQUED = aqNaq or aqCaq, R is not accessed. R is an input argument if FACT = aqFaq; otherwise, R is an output argument. If FACT = aqFaq and EQUED = aqRaq or aqBaq, 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.
 C (input or output) REAL array, dimension (N)
 The column scale factors for A. If EQUED = aqCaq or aqBaq, A is multiplied on the right by diag(C); if EQUED = aqNaq or aqRaq, C is not accessed. C is an input argument if FACT = aqFaq; otherwise, C is an output argument. If FACT = aqFaq and EQUED = aqCaq or aqBaq, 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.
 B (input/output) REAL array, dimension (LDB,NRHS)
 On entry, the NbyNRHS right hand side matrix B. On exit, if EQUED = aqNaq, B is not modified; if TRANS = aqNaq and EQUED = aqRaq or aqBaq, B is overwritten by diag(R)*B; if TRANS = aqTaq or aqCaq and EQUED = aqCaq or aqBaq, B is overwritten by diag(C)*B.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 X (output) REAL array, dimension (LDX,NRHS)
 If INFO = 0, the NbyNRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. aqNaq, and the solution to the equilibrated system is inv(diag(C))*X if TRANS = aqNaq and EQUED = aqCaq or aqBaq, or inv(diag(R))*X if TRANS = aqTaq or aqCaq and EQUED = aqRaq or aqBaq.
 LDX (input) INTEGER
 The leading dimension of the array X. LDX >= max(1,N).
 RCOND (output) 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.
 RPVGRW (output) 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<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).
 BERR (output) 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). N_ERR_BNDS (input) 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.
 ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)
 For each righthand 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 righthand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/donaqt trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch(aqEpsilonaq). 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(aqEpsilonaq). 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(aqEpsilonaq) 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.
 ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
 For each righthand 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 righthand 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 righthand 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 righthand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/donaqt trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch(aqEpsilonaq). 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(aqEpsilonaq). 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(aqEpsilonaq) 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 righthand 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. NPARAMS (input) INTEGER Specifies the number of parameters set in PARAMS. If .LE. 0, the PARAMS array is never referenced and default values are used.
 PARAMS (input / output) 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 highernumbered 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 doubleprecision refinement algorithm, possibly with doubledsingle 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 doubleprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)  WORK (workspace) REAL array, dimension (4*N)
 IWORK (workspace) INTEGER array, dimension (N)
 INFO (output) INTEGER

= 0: Successful exit. The solution to every righthand side is guaranteed. < 0: If INFO = i, the ith 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 righthand 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 righthand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth righthand 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 righthand 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 righthand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.