F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07JAF (DPTSV)

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

F07JAF (DPTSV) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric positive definite tridiagonal matrix, and $X$ and $B$ are $n$ by $r$ matrices.

## 2  Specification

 SUBROUTINE F07JAF ( N, NRHS, D, E, B, LDB, INFO)
 INTEGER N, NRHS, LDB, INFO REAL (KIND=nag_wp) D(*), E(*), B(LDB,*)
The routine may be called by its LAPACK name dptsv.

## 3  Description

F07JAF (DPTSV) factors $A$ as $A=LD{L}^{\mathrm{T}}$. The factored form of $A$ is then used to solve the system of equations.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
3:     D($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ diagonal elements of the tridiagonal matrix $A$.
On exit: the $n$ diagonal elements of the diagonal matrix $D$ from the factorization $A=LD{L}^{\mathrm{T}}$.
4:     E($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array E must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: the $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
On exit: the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$. (E can also be regarded as the superdiagonal of the unit bidiagonal factor $U$ from the ${U}^{\mathrm{T}}DU$ factorization of $A$.)
5:     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: if ${\mathbf{INFO}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
6:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07JAF (DPTSV) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
7:     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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the leading minor of order $i$ is not positive definite, and the solution has not been computed. The factorization has not been completed unless $i={\mathbf{N}}$.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
F07JBF (DPTSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BGF solves $Ax=b$ and returns a forward error bound and condition estimate. F04BGF calls F07JAF (DPTSV) to solve the equations.

The number of floating point operations required for the factorization of $A$ is proportional to $n$, and the number of floating point operations required for the solution of the equations is proportional to $nr$, where $r$ is the number of right-hand sides.
The complex analogue of this routine is F07JNF (ZPTSV).

## 9  Example

This example solves the equations
 $Ax=b ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 and b = 6.0 9.0 2.0 14.0 7.0 .$
Details of the $LD{L}^{\mathrm{T}}$ factorization of $A$ are also output.

### 9.1  Program Text

Program Text (f07jafe.f90)

### 9.2  Program Data

Program Data (f07jafe.d)

### 9.3  Program Results

Program Results (f07jafe.r)