f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dtptri (f07ujc)

## 1  Purpose

nag_dtptri (f07ujc) computes the inverse of a real triangular matrix, using packed storage.

## 2  Specification

 #include #include
 void nag_dtptri (Nag_OrderType order, Nag_UploType uplo, Nag_DiagType diag, Integer n, double ap[], NagError *fail)

## 3  Description

nag_dtptri (f07ujc) forms the inverse of a real triangular matrix $A$, using packed storage. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

## 4  References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     diagNag_DiagTypeInput
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or $\mathrm{Nag_UnitDiag}$.
4:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     ap[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ triangular matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
If ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$, the diagonal elements of $\mathrm{AP}$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$.
On exit: $A$ is overwritten by ${A}^{-1}$, using the same storage format as described above.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
$a\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$ is exactly zero. $A$ is singular its inverse cannot be computed.

## 7  Accuracy

The computed inverse $X$ satisfies
 $XA-I≤cnεXA ,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
Note that a similar bound for $\left|AX-I\right|$ cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
 $X-A-1≤cnεA-1AX .$
See Du Croz and Higham (1992).

## 8  Parallelism and Performance

nag_dtptri (f07ujc) is not threaded by NAG in any implementation.
nag_dtptri (f07ujc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

The total number of floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
The complex analogue of this function is nag_ztptri (f07uwc).

## 10  Example

This example computes the inverse of the matrix $A$, where
 $A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12 ,$
using packed storage.

### 10.1  Program Text

Program Text (f07ujce.c)

### 10.2  Program Data

Program Data (f07ujce.d)

### 10.3  Program Results

Program Results (f07ujce.r)