# NAG Library Routine Document

## 1Purpose

f07jef (dpttrs) 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, using the $LD{L}^{\mathrm{T}}$ factorization returned by f07jdf (dpttrf).

## 2Specification

Fortran Interface
 Subroutine f07jef ( n, nrhs, d, e, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*), e(*) Real (Kind=nag_wp), Intent (Inout) :: b(ldb,*)
#include nagmk26.h
 void f07jef_ (const Integer *n, const Integer *nrhs, const double d[], const double e[], double b[], const Integer *ldb, Integer *info)
The routine may be called by its LAPACK name dpttrs.

## 3Description

f07jef (dpttrs) should be preceded by a call to f07jdf (dpttrf), which computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLT ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. f07jef (dpttrs) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form ${U}^{\mathrm{T}}DU$, where $U$ is a unit upper bidiagonal matrix.

## 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 http://www.netlib.org/lapack/lug

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{nrhs}$ – 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:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
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: must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
4:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
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: must contain the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix $U$ from the ${U}^{\mathrm{T}}DU$ factorization of $A$.)
5:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $r$ matrix of right-hand sides $B$.
On exit: the $n$ by $r$ solution matrix $X$.
6:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07jef (dpttrs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{info}$ – IntegerOutput
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

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 =OεA1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^ - x 1 x1 ≤ κA E1 A1 ,$
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.
Following the use of this routine f07jgf (dptcon) can be used to estimate the condition number of $A$ and f07jhf (dptrfs) can be used to obtain approximate error bounds.

## 8Parallelism and Performance

f07jef (dpttrs) 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 required to solve the equations $AX=B$ is proportional to $nr$.
The complex analogue of this routine is f07jsf (zpttrs).

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = 4.0 -2.0 0.0 0.0 0.0 -2.0 10.0 -6.0 0.0 0.0 0.0 -6.0 29.0 15.0 0.0 0.0 0.0 15.0 25.0 8.0 0.0 0.0 0.0 8.0 5.0 and B = 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .$

### 10.1Program Text

Program Text (f07jefe.f90)

### 10.2Program Data

Program Data (f07jefe.d)

### 10.3Program Results

Program Results (f07jefe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017