# ?tptri

Computes the inverse of a triangular matrix using packed storage.

## Syntax

FORTRAN 77:

call stptri( uplo, diag, n, ap, info )

call dtptri( uplo, diag, n, ap, info )

call ctptri( uplo, diag, n, ap, info )

call ztptri( uplo, diag, n, ap, info )

Fortran 95:

call tptri( a [,uplo] [,diag] [,info] )

## Description

This routine computes the inverse `inv(A)` of a packed triangular matrix A.

## 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.

diag

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

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

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

n

  INTEGER. The order of the matrix A; `n ≥ 0`.

ap

  REAL for stptri

  DOUBLE PRECISION for dtptri

  COMPLEX for ctptri

  DOUBLE COMPLEX for ztptri.

  Array, DIMENSION at least `max(1,n(n+1)/2)`.

  Contains the packed triangular matrix A.

## Output Parameters

ap

  Overwritten by the packed n-by-n matrix `inv(A)` .

info

  INTEGER.

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

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

  If `info = i`, the i-th diagonal element of A is zero, A is singular, and the inversion could not be completed.

## 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 tptri interface are as follows:

a

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

uplo

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

diag

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

## Application Notes

The computed inverse X satisfies the following error bounds:
`|XA - I| ≤ c(n)ε |X||A|`
`|X - A-1| ≤ c(n)ε |A-1||A||X|,`

where `c(n)` is a modest linear function of n; ε is the machine precision; I denotes the identity matrix.

The total number of floating-point operations is approximately `(1/3)n3` for real flavors and `(4/3)n3` for complex flavors.

Parent topic: Routines for Matrix Inversion