# NAG FL Interfacef07jaf (dptsv)

## 1Purpose

f07jaf 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.

## 2Specification

Fortran Interface
 Subroutine f07jaf ( n, nrhs, d, e, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), b(ldb,*)
#include <nag.h>
 void f07jaf_ (const Integer *n, const Integer *nrhs, double d[], double e[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07jaf, nagf_lapacklin_dptsv or its LAPACK name dptsv.

## 3Description

f07jaf factors $A$ as $A=LD{L}^{\mathrm{T}}$. The factored form of $A$ is then used to solve the system of equations.

## 4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{nrhs}$Integer Input
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: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input/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: $\mathbf{e}\left(*\right)$Real (Kind=nag_wp) array Input/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: $\mathbf{b}\left({\mathbf{ldb}},*\right)$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 $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
6: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07jaf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7: $\mathbf{info}$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.
${\mathbf{info}}>0$
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite, and the solution has not been computed. The factorization has not been completed unless ${\mathbf{n}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

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 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 to solve the equations.

## 8Parallelism and Performance

f07jaf 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 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.

## 10Example

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.

### 10.1Program Text

Program Text (f07jafe.f90)

### 10.2Program Data

Program Data (f07jafe.d)

### 10.3Program Results

Program Results (f07jafe.r)