SPTSVX(3f)

NAME
SPTSVX - 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. 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.

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 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      (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 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       (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 of INFO = N+1, 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 particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.

FERR    (output) 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    (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, 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.