dsysvxx.f man page

dsysvxx.f —

Synopsis

Functions/Subroutines

subroutine dsysvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
DSYSVXX

Function/Subroutine Documentation

subroutine dsysvxx (characterFACT, characterUPLO, integerN, integerNRHS, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, characterEQUED, double precision, dimension( * )S, 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)

DSYSVXX

Purpose:

DSYSVXX uses the diagonal pivoting factorization to compute the
solution to a double precision system of linear equations A * X = B, where A
is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds
are returned. DSYSVXX 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.

DSYSVXX 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
DSYSVXX 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 DSYSVXX would itself produce.

Description:

The following steps are performed:

1. If FACT = 'E', double precision scaling factors are computed to equilibrate
the system:

  diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B

Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the LU decomposition is used to factor
the matrix A (after equilibration if FACT = 'E') as

   A = U * D * U**T,  if UPLO = 'U', or
   A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

3. If some D(i,i)=0, so that D is exactly singular, then the
routine returns with INFO = i. Otherwise, the factored form of A
is used to estimate the condition number of the matrix A (see
argument RCOND).  If the reciprocal of the condition number is
less than machine precision, the routine still goes on to solve
for X and compute error bounds as described below.

4. The system of equations is solved for X using the factored form
of A.

5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds.  Refinement calculates the residual to at
least twice the working precision.

6. If equilibration was used, the matrix X is premultiplied by
diag(R) so that it solves the original system before
equilibration.
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 S.
          A, AF, and IPIV are not modified.
  = 'N':  The matrix A will be copied to AF and factored.
  = 'E':  The matrix A will be equilibrated if necessary, then
          copied to AF and factored.

UPLO

   UPLO is CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N

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

A

     A is DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular
part of the matrix A, and the strictly lower triangular
part of A is not referenced.  If UPLO = 'L', the leading
N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).

LDA

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

AF

     AF is DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers
used to obtain the factor U or L from the factorization A =
U*D*U**T or A = L*D*L**T as computed by DSYTRF.

If FACT = 'N', then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers
used to obtain the factor U or L from the factorization A =
U*D*U**T or A = L*D*L**T.

LDAF

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

IPIV

     IPIV is INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block
structure of D, as determined by DSYTRF.  If IPIV(k) > 0,
then rows and columns k and IPIV(k) were interchanged and
D(k,k) is a 1-by-1 diagonal block.  If UPLO = 'U' and
IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
-IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
then rows and columns k+1 and -IPIV(k) were interchanged
and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block
structure of D, as determined by DSYTRF.

EQUED

     EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
  = 'N':  No equilibration (always true if FACT = 'N').
  = 'Y':  Both row and column equilibration, i.e., A has been
          replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.

S

     S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.  If EQUED = 'Y', A is multiplied on
the left and right by diag(S).  S is an input argument if FACT =
'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
= 'Y', each element of S must be positive.  If S is output, each
element of S is a power of the radix. If S is input, each element
of S should be a power of the radix to ensure a reliable solution
and error estimates. Scaling by powers of the radix does not cause
rounding errors unless the result underflows or overflows.
Rounding errors during scaling lead to refining with a matrix that
is not equivalent to the input matrix, producing error estimates
that may not be reliable.

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 EQUED = 'Y', B is overwritten by diag(S)*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(S))*X.

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.

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:

September 2012

Definition at line 503 of file dsysvxx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

dsysvxx(3) is an alias of dsysvxx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK