# NAG FL Interfacef07wef (dpftrs)

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## 1Purpose

f07wef solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
using the Cholesky factorization computed by f07wdf stored in Rectangular Full Packed (RFP) format.

## 2Specification

Fortran Interface
 Subroutine f07wef ( uplo, n, nrhs, ar, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: ar(n*(n+1)/2) Real (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: transr, uplo
#include <nag.h>
 void f07wef_ (const char *transr, const char *uplo, const Integer *n, const Integer *nrhs, const double ar[], double b[], const Integer *ldb, Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by the names f07wef, nagf_lapacklin_dpftrs or its LAPACK name dpftrs.

## 3Description

f07wef is used to solve a real symmetric positive definite system of linear equations $AX=B$, the routine must be preceded by a call to f07wdf which computes the Cholesky factorization of $A$, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{T}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{T}}X=Y$.

## 4References

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: $\mathbf{transr}$Character(1) Input
On entry: specifies whether the RFP representation of $A$ is normal or transposed.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'T'}$
The matrix $A$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'T'}$.
2: $\mathbf{uplo}$Character(1) Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$Real (Kind=nag_wp) array Input
On entry: the Cholesky factorization of $A$ stored in RFP format, as returned by f07wdf.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
On exit: the $n×r$ solution matrix $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07wef is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{'U'}$, $|E|\le c\left(n\right)\epsilon |{U}^{\mathrm{T}}||U|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $|E|\le c\left(n\right)\epsilon |L||{L}^{\mathrm{T}}|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where $\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ and ${\kappa }_{\infty }\left(A\right)$ is the condition number when using the $\infty$-norm.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.

## 8Parallelism and Performance

f07wef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $2{n}^{2}r$.
The complex analogue of this routine is f07wsf.

## 10Example

This example solves the system of equations $AX=B$, where
 $A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) and B= ( 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 ) .$
Here $A$ is symmetric positive definite, stored in RFP format, and must first be factorized by f07wdf.

### 10.1Program Text

Program Text (f07wefe.f90)

### 10.2Program Data

Program Data (f07wefe.d)

### 10.3Program Results

Program Results (f07wefe.r)