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# sptsvx

```
NAME
SPTSVX - use 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

SYNOPSIS
SUBROUTINE SPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, INFO )

CHARACTER      FACT

INTEGER        INFO, LDB, LDX, N, NRHS

REAL           RCOND

REAL           B( LDB, * ), BERR( * ), D( * ), DF( * ),
E( * ), EF( * ), FERR( * ), WORK( * ), X(
LDX, * )

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. The factored form of A is used to compute the condition
number
of the matrix A.  If the reciprocal of the condition
number is
less than machine precision, steps 3 and 4 are skipped.

3. The system of equations is solved for X using the fac-
tored form
of A.

4. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error

estimates
for it.

ARGUMENTS
FACT    (input) 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       (input) INTEGER
The order of the matrix A.  N >= 0.

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

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

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

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

EF      (input or output) 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 factori-
zation 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       (input) REAL array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix 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 N-by-NRHS solution matrix X.

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

RCOND   (output) REAL
The reciprocal condition number of the matrix A.  If
RCOND is less than the machine precision (in partic-
ular, if RCOND = 0), the matrix is singular to work-
ing precision.  This condition is indicated by a
return code of INFO > 0, and the solution and error
bounds are not computed.

FERR    (output) REAL array, dimension (NRHS)
The estimated forward error bounds 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) bounds the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude
of the largest element in X(j).

BERR    (output) 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    (workspace) REAL array, dimension (2*N)

INFO    (output) 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 unless i = N,
and the solution and error bounds could not be com-
puted.  = N+1 RCOND is less than machine precision.
The factorization has been completed, but the matrix
is singular to working precision, and the solution
and error bounds have not been computed.
```