F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HEF (DPBTRS)

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

F07HEF (DPBTRS) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by F07HDF (DPBTRF).

## 2  Specification

 SUBROUTINE F07HEF ( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
 INTEGER N, KD, NRHS, LDAB, LDB, INFO REAL (KIND=nag_wp) AB(LDAB,*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpbtrs.

## 3  Description

F07HEF (DPBTRS) is used to solve a real symmetric positive definite band system of linear equations $AX=B$, the routine must be preceded by a call to F07HDF (DPBTRF) which computes the Cholesky factorization of $A$. 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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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 $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{KD}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     AB(LDAB,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the Cholesky factor of $A$, as returned by F07HDF (DPBTRF).
6:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HEF (DPBTRS) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
7:     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 $X$.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07HEF (DPBTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     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 ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(k+1\right)$ is a modest linear function of $k+1$, 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
 $x-x^∞ x∞ ≤ck+1condA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling F07HHF (DPBRFS), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling F07HGF (DPBCON).

The total number of floating point operations is approximately $4nkr$, assuming $n\gg k$.
This routine may be followed by a call to F07HHF (DPBRFS) to refine the solution and return an error estimate.
The complex analogue of this routine is F07HSF (ZPBTRS).

## 9  Example

This example solves the system of equations $AX=B$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 and B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .$
Here $A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by F07HDF (DPBTRF).

### 9.1  Program Text

Program Text (f07hefe.f90)

### 9.2  Program Data

Program Data (f07hefe.d)

### 9.3  Program Results

Program Results (f07hefe.r)