F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07WEF (DPFTRS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07WEF (DPFTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by F07WDF (DPFTRF), stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

## 2  Specification

 SUBROUTINE F07WEF ( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
 INTEGER N, NRHS, LDB, INFO REAL (KIND=nag_wp) A(N*(N+1)/2), B(LDB,*) CHARACTER(1) TRANSR, UPLO
The routine may be called by its LAPACK name dpftrs.

## 3  Description

F07WEF (DPFTRS) 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 (DPFTRF) which computes the Cholesky factorization of $A$, where $A$ is stored in RFP format. 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 ${L}^{\mathrm{T}}X=Y$.

## 4  References

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

## 5  Parameters

1:     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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     A(${\mathbf{N}}×\left({\mathbf{N}}+1\right)/2$) – REAL (KIND=nag_wp) arrayInput
On entry: the Cholesky factorization of $A$ stored in RFP format, as returned by F07WDF (DPFTRF).
6:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/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 $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07WEF (DPFTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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 ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(n\right)$ 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
 $x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ 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 $\mathrm{cond}\left(A\right)$.

The total number of floating point operations is approximately $2{n}^{2}r$.
The complex analogue of this routine is F07WSF (ZPFTRS).

## 9  Example

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 (DPFTRF).

### 9.1  Program Text

Program Text (f07wefe.f90)

### 9.2  Program Data

Program Data (f07wefe.d)

### 9.3  Program Results

Program Results (f07wefe.r)