# NAG CL Interfacef07wxc (ztftri)

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## 1Purpose

f07wxc computes the inverse of a complex triangular matrix stored in Rectangular Full Packed (RFP) format.

## 2Specification

 #include
 void f07wxc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Nag_DiagType diag, Integer n, Complex ar[], NagError *fail)
The function may be called by the names: f07wxc, nag_lapacklin_ztftri or nag_ztftri.

## 3Description

f07wxc forms the inverse of a complex triangular matrix $A$, stored using RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

## 4References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{transr}$Nag_RFP_Store Input
On entry: specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\mathrm{Nag_RFP_ConjTrans}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$ or $\mathrm{Nag_RFP_ConjTrans}$.
3: $\mathbf{uplo}$Nag_UploType Input
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 $\mathrm{Nag_Lower}$.
4: $\mathbf{diag}$Nag_DiagType Input
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 $\mathrm{Nag_UnitDiag}$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{ar}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$Complex Input/Output
On entry: the upper or lower triangular part (as specified by uplo) of the $n×n$ Hermitian matrix $A$, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.4.3 in the F07 Chapter Introduction.
On exit: $A$ is overwritten by ${A}^{-1}$, in the same storage format as $A$.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
Diagonal element $⟨\mathit{\text{value}}⟩$ of $A$ is exactly zero. $A$ is singular its inverse cannot be computed.

## 7Accuracy

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

## 8Parallelism and Performance

f07wxc 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.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
The real analogue of this function is f07wkc.

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= ( 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i )$
and is stored using RFP format.

### 10.1Program Text

Program Text (f07wxce.c)

### 10.2Program Data

Program Data (f07wxce.d)

### 10.3Program Results

Program Results (f07wxce.r)