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realPTsolve(3) LAPACK realPTsolve(3)

NAME

realPTsolve - real

SYNOPSIS

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.

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.

Author

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