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dgbsvxx-f(3)

dgbsvxx.f –

Synopsis

Functions/Subroutines

subroutine dgbsvxx (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)
DGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Function/Subroutine Documentation

subroutine dgbsvxx (characterFACT, characterTRANS, integerN, integerKL, integerKU, integerNRHS, double precision, dimension( ldab, * )AB, integerLDAB, double precision, dimension( ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, characterEQUED, double precision, dimension( * )R, double precision, dimension( * )C, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( ldx , * )X, integerLDX, double precisionRCOND, double precisionRPVGRW, double precision, dimension( * )BERR, integerN_ERR_BNDS, double precision, dimension( nrhs, * )ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, integerNPARAMS, double precision, dimension( * )PARAMS, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

DGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

 DGBSVXX uses the LU factorization to compute the solution to a
 double precision 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. DGBSVXX 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.
 DGBSVXX 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
 DGBSVXX 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 DGBSVXX would itself produce.

 

Description:

 The following steps are performed:
 1. If FACT = 'E', double precision 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.

Parameters:

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.

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)

N

 N is INTEGER
 The number of linear equations, i.e., the order of the
 matrix A. N >= 0.

KL

 KL is INTEGER
 The number of subdiagonals within the band of A. KL >= 0.

KU

 KU is INTEGER
 The number of superdiagonals within the band of A. KU >= 0.

NRHS

 NRHS is INTEGER
 The number of right hand sides, i.e., the number of columns
 of the matrices B and X. NRHS >= 0.

AB

 AB is DOUBLE PRECISION 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).

LDAB

 LDAB is INTEGER
 The leading dimension of the array AB. LDAB >= KL+KU+1.

AFB

 AFB is DOUBLE PRECISION 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 DGBTRF. 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).

LDAFB

 LDAFB is INTEGER
 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

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 DGETRF; 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.

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.

R

 R is DOUBLE PRECISION 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.

C

 C is DOUBLE PRECISION 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.

B

 B is DOUBLE PRECISION 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.

LDB

 LDB is INTEGER
 The leading dimension of the array B. LDB >= max(1,N).

X

 X is DOUBLE PRECISION 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'.

LDX

 LDX is INTEGER
 The leading dimension of the array X. LDX >= max(1,N).

RCOND

 RCOND is DOUBLE PRECISION
 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

 RPVGRW is DOUBLE PRECISION
 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 DGESVX, this quantity is
 returned in WORK(1).

BERR

 BERR is DOUBLE PRECISION 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

 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.

ERR_BNDS_NORM

 ERR_BNDS_NORM is DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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.

ERR_BNDS_COMP

 ERR_BNDS_COMP is DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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.

NPARAMS

 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.

PARAMS

 PARAMS is DOUBLE PRECISION 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: 1.0D+0
 = 0.0 : No refinement is performed, and no error bounds are
 computed.
 = 1.0 : Use the extra-precise refinement algorithm.
 (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)

WORK

 WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

 IWORK is INTEGER array, dimension (N)

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.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Definition at line 557 of file dgbsvxx.f.

Author

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