# ?tptrs

Solves a system of linear equations with a packed triangular matrix, with multiple right-hand sides.

## Syntax

FORTRAN 77:

call stptrs( uplo, trans, diag, n, nrhs, ap, b, ldb, info )

call dtptrs( uplo, trans, diag, n, nrhs, ap, b, ldb, info )

call ctptrs( uplo, trans, diag, n, nrhs, ap, b, ldb, info )

call ztptrs( uplo, trans, diag, n, nrhs, ap, b, ldb, info )

Fortran 95:

call tptrs( a, b [,uplo] [, trans] [,diag] [,info] )

## Description

This routine solves for X the following systems of linear equations with a packed triangular matrix A, with multiple right-hand sides stored in B:

A*X = B

  if trans='N',

`AT*X = B`

  if trans='T',

AH*X = B

  if trans='C' (for complex matrices only).

## Input Parameters

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates whether A is upper or lower triangular:

  If `uplo = 'U'`, then A is upper triangular.

  If `uplo = 'L'`, then A is lower triangular.

trans

  CHARACTER*1. Must be 'N' or 'T' or 'C'.

  If `trans = 'N'`, then `A*X = B` is solved for X.

  If `trans = 'T'`, then AT*X = B is solved for X.

  If `trans = 'C'`, then AH*X = B is solved for X.

diag

  CHARACTER*1. Must be 'N' or 'U'.

  If `diag = 'N'`, then A is not a unit triangular matrix.

  If `diag = 'U'`, then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ap.

n

  INTEGER. The order of A; the number of rows in B; n ≥ 0.

nrhs

  INTEGER. The number of right-hand sides; nrhs ≥ 0.

ap, b

  REAL for stptrs

  DOUBLE PRECISION for dtptrs

  COMPLEX for ctptrs

  DOUBLE COMPLEX for ztptrs.

  Arrays: `ap(*)`, `b(ldb,*)`.

  The dimension of ap must be at least max(1,n(n+1)/2). The array ap contains the matrix A in packed storage (see Matrix Storage Schemes).

  The array b contains the matrix B whose columns are the right-hand sides for the system of equations. The second dimension of b must be at least `max(1, nrhs)`.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

## Output Parameters

b

  Overwritten by the solution matrix X.

info

  INTEGER. If `info=0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine tptrs interface are as follows:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size `(n*(n+1)/2)`.

b

  Holds the matrix B of size (n, nrhs).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

trans

  Must be 'N', 'C', or 'T'. The default value is 'N'.

diag

  Must be 'N' or 'U'. The default value is 'N'.

## Application Notes

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations `(A + E)x = b`, where
`|E| ≤ c(n)ε |A|`

`c(n)` is a modest linear function of n, and ε is the machine precision.

If `x0` is the true solution, the computed solution x satisfies this error bound:

where `cond(A,x)`= || |A-1||A| |x| ||∞ / ||x||∞ ≤ ||A-1||∞ ||A||∞ = κ∞(A).

Note that `cond(A,x)` can be much smaller than `κ∞(A)`; the condition number of `AT` and `AH` might or might not be equal to `κ∞(A)`.

The approximate number of floating-point operations for one right-hand side vector b is n2 for real flavors and `4n2` for complex flavors.

To estimate the condition number `κ∞(A)`, call ?tpcon.

To estimate the error in the solution, call ?tprfs.

Parent topic: Routines for Solving Systems of Linear Equations