?ptsvx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?ptsvx

Uses factorization to compute the solution to the system of linear equations with a symmetric (Hermitian) positive definite tridiagonal coefficient matrix A, and provides error bounds on the solution.

Syntax

call sptsvx( fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, info )

call dptsvx( fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, info )

call cptsvx( fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call zptsvx( fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call ptsvx( d, e, b, x [,df] [,ef] [,fact] [,ferr] [,berr] [,rcond] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine uses the Cholesky factorization A = L*D*L^T (real)/A = L*D*L^H (complex) to compute the solution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric or Hermitian positive definite tridiagonal matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

Error bounds on the solution and a condition estimate are also provided.

The routine ?ptsvx performs the following steps:

If fact = 'N', the matrix A is factored as A = L*D*L^T (real flavors)/A = L*D*L^H (complex flavors), 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 (real flavors)/A = U^H*D*U (complex flavors).
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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Input Parameters

fact	CHARACTER*1. Must be 'F' or 'N'. Specifies whether or not the factored form of the matrix `A` is supplied on entry. If `fact = 'F'`: on entry, `df` and `ef` contain the factored form of `A`. Arrays `d`, `e`, `df`, and `ef` will not be modified. If `fact = 'N'`, the matrix `A` will be copied to `df` and `ef`, and factored.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
nrhs	INTEGER. The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
d, df, rwork	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays: `d` (size `n`), `df` (size `n`), `rwork(n)`. The array `d` contains the `n` diagonal elements of the tridiagonal matrix `A`. The array `df` is an input argument if `fact = 'F'` and on entry contains the `n` diagonal elements of the diagonal matrix `D` from the `LDL^T` (real)/ `LDL^H` (complex) factorization of `A`. The array `rwork` is a workspace array used for complex flavors only.
e,ef,b,work	REAL for sptsvx DOUBLE PRECISION for dptsvx COMPLEX for cptsvx DOUBLE COMPLEX for zptsvx. Arrays: `e` (size `n` -1), `ef` (size `n` -1), `b`(size ldb, ), `work()`. The array `e` contains the `(n - 1)` subdiagonal elements of the tridiagonal matrix `A`. The array `ef` is an input argument if `fact = 'F'` and on entry contains the `(n - 1)` subdiagonal elements of the unit bidiagonal factor `L` from the `LDL^T` (real)/ `LDL^H` (complex) factorization of `A`. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The array `work` is a workspace array. The dimension of `work` must be at least `2*n` for real flavors, and at least `n` for complex flavors.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.
ldx	INTEGER. The leading dimension of `x`; `ldx≥ max(1, n)`.

Output Parameters

x	REAL for sptsvx DOUBLE PRECISION for dptsvx COMPLEX for cptsvx DOUBLE COMPLEX for zptsvx. Array, size `ldx` by *. If `info = 0` or `info = n+1`, the array `x` contains the solution matrix `X` to the system of equations. The second dimension of `x` must be at least `max(1,nrhs)`.
df, ef	These arrays are output arguments if `fact = 'N'`. See the description of `df`, `ef` in Input Arguments section.
rcond	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. An estimate of the reciprocal condition number of the matrix `A` after equilibration (if done). 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains the estimated 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)`. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
berr	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains the component-wise relative backward error for each solution vector `x(j)`, that is, the smallest relative change in any element of `A` or `B` that makes `x(j)` an exact solution.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, and `i≤n`, the leading minor of order `i` (and therefore the matrix `A` itself) is not positive-definite, so the factorization could not be completed, and the solution and error bounds could not be computed; `rcond` =0 is returned. If `info = i`, and `i` = `n` + 1, then `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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine ptsvx interface are as follows:

d	Holds the vector of length `n`.
e	Holds the vector of length (`n-1`).
b	Holds the matrix `B` of size (`n,nrhs`).
x	Holds the matrix `X` of size (`n,nrhs`).
df	Holds the vector of length `n`.
ef	Holds the vector of length (`n-1`).
ferr	Holds the vector of length (`nrhs`).
berr	Holds the vector of length (`nrhs`).
fact	Must be 'N' or 'F'. The default value is 'N'. If `fact = 'F'`, then both arguments `af` and `ipiv` must be present; otherwise, an error is returned.

Parent topic: LAPACK Linear Equation Driver Routines

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