Functions

subroutine sptsv (N, NRHS, D, E, B, LDB, INFO)
SPTSV computes the solution to system of linear equations A * X = B for PT matrices
subroutine sptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO)
SPTSVX computes the solution to system of linear equations A * X = B for PT matrices

Detailed Description

This is the group of real solve driver functions for PT matrices

Function Documentation

subroutine sptsv (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPTSV computes the solution to system of linear equations A * X = B for PT matrices

Purpose:

 SPTSV computes the solution to a real system of linear equations
 A*X = B, where A is an N-by-N symmetric positive definite tridiagonal
 matrix, and X and B are N-by-NRHS matrices.
 A is factored as A = L*D*L**T, and the factored form of A is then
 used to solve the system of equations.

 

Parameters:

N

          N is INTEGER
          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 matrix B.  NRHS >= 0.

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the factorization A = L*D*L**T.

E

          E is REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L**T factorization of
          A.  (E can also be regarded as the superdiagonal of the unit
          bidiagonal factor U from the U**T*D*U factorization of A.)

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.

LDB

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

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the solution has not been
                computed.  The factorization has not been completed
                unless i = N.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sptsvx (character FACT, integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)

SPTSVX computes the solution to system of linear equations A * X = B for PT matrices

Purpose:

 SPTSVX uses the factorization A = L*D*L**T to compute the solution
 to a real system of linear equations A*X = B, where A is an N-by-N
 symmetric positive definite tridiagonal matrix and X and B are
 N-by-NRHS matrices.
 Error bounds on the solution and a condition estimate are also
 provided.

 

Description:

 The following steps are performed:
 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
    is a unit lower bidiagonal matrix and D is diagonal.  The
    factorization can also be regarded as having the form
    A = U**T*D*U.
 2. If the leading i-by-i principal minor is not positive definite,
    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.
 3. The system of equations is solved for X using the factored form
    of A.
 4. Iterative refinement is applied to improve the computed solution
    matrix and calculate error bounds and backward error estimates
    for it.

 

Parameters:

FACT

          FACT is CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, DF and EF contain the factored form of A.
                  D, E, DF, and EF will not be modified.
          = 'N':  The matrix A will be copied to DF and EF and
                  factored.

N

          N is INTEGER
          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.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.

E

          E is REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

DF

          DF is REAL array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.
          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the diagonal matrix D
          from the L*D*L**T factorization of A.

EF

          EF is REAL array, dimension (N-1)
          If FACT = 'F', then EF is an input argument and on entry
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.
          If FACT = 'N', then EF is an output argument and on exit
          contains the (n-1) subdiagonal elements of the unit
          bidiagonal factor L from the L*D*L**T factorization of A.

B

          B is REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

LDB

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

X

          X is REAL array, dimension (LDX,NRHS)
          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

          RCOND is REAL
          The reciprocal condition number of the matrix A.  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.

FERR

          FERR is REAL array, dimension (NRHS)
          The 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).

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).

WORK

          WORK is REAL array, dimension (2*N)

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:  the leading minor of order i of A is
                       not positive definite, so the factorization
                       could not be completed, and the solution has not
                       been 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
                       value of RCOND would suggest.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Author

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