# NAG CL Interfacef07wjc (dpftri)

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## 1Purpose

f07wjc computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by f07wdc stored in Rectangular Full Packed (RFP) format.

## 2Specification

 #include
 void f07wjc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, double ar[], NagError *fail)
The function may be called by the names: f07wjc, nag_lapacklin_dpftri or nag_dpftri.

## 3Description

f07wjc is used to compute the inverse of a real symmetric positive definite matrix $A$, stored in RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction. The function must be preceded by a call to f07wdc, which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $A={U}^{\mathrm{T}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $A=L{L}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

## 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 RFP representation of $A$ is normal or transposed.
${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\mathrm{Nag_RFP_Trans}$
The matrix $A$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$ or $\mathrm{Nag_RFP_Trans}$.
3: $\mathbf{uplo}$Nag_UploType Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{ar}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$double Input/Output
On entry: the Cholesky factorization of $A$ stored in RFP format, as returned by f07wdc.
On exit: the factorization is overwritten by the $n×n$ matrix ${A}^{-1}$ stored in RFP format.
6: $\mathbf{fail}$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_MAT_NOT_POS_DEF
The leading minor of order $⟨\mathit{\text{value}}⟩$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no function specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling f07mjc.
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.

## 7Accuracy

The computed inverse $X$ satisfies
 $‖XA-I‖2≤c(n)εκ2(A) and ‖AX-I‖2≤c(n)εκ2(A) ,$
where $c\left(n\right)$ is a modest function of $n$, $\epsilon$ is the machine precision and ${\kappa }_{2}\left(A\right)$ is the condition number of $A$ defined by
 $κ2(A)=‖A‖2‖A-1‖2 .$

## 8Parallelism and Performance

f07wjc 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 floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogue of this function is f07wwc.

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) .$
Here $A$ is symmetric positive definite, stored in RFP format, and must first be factorized by f07wdc.

### 10.1Program Text

Program Text (f07wjce.c)

### 10.2Program Data

Program Data (f07wjce.d)

### 10.3Program Results

Program Results (f07wjce.r)