F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07JSF (ZPTTRS)

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

F07JSF (ZPTTRS) computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ Hermitian positive definite tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices, using the $LD{L}^{\mathrm{H}}$ factorization returned by F07JRF (ZPTTRF).

## 2  Specification

 SUBROUTINE F07JSF ( UPLO, N, NRHS, D, E, B, LDB, INFO)
 INTEGER N, NRHS, LDB, INFO REAL (KIND=nag_wp) D(*) COMPLEX (KIND=nag_wp) E(*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zpttrs.

## 3  Description

F07JSF (ZPTTRS) should be preceded by a call to F07JRF (ZPTTRF), which computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLH ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. F07JSF (ZPTTRS) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form ${U}^{\mathrm{H}}DU$, where $U$ is a unit upper bidiagonal matrix.

## 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

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies the form of the factorization as follows:
${\mathbf{UPLO}}=\text{'U'}$
$A={U}^{\mathrm{H}}DU$.
${\mathbf{UPLO}}=\text{'L'}$
$A=LD{L}^{\mathrm{H}}$.
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:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     D($*$) – 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 $LD{L}^{\mathrm{H}}$ or ${U}^{\mathrm{H}}DU$ factorization of $A$.
5:     E($*$) – COMPLEX (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: if ${\mathbf{UPLO}}=\text{'U'}$, E must contain the $\left(n-1\right)$ superdiagonal elements of the unit upper bidiagonal matrix $U$ from the ${U}^{\mathrm{H}}DU$ factorization of $A$.
If ${\mathbf{UPLO}}=\text{'L'}$, E must contain the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$.
6:     B(LDB,$*$) – COMPLEX (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 $X$.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07JSF (ZPTTRS) 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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 =OεA1$
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 F07JUF (ZPTCON) can be used to estimate the condition number of $A$ and F07JVF (ZPTRFS) can be used to obtain approximate error bounds.

The total number of floating point operations required to solve the equations $AX=B$ is proportional to $nr$.
The real analogue of this routine is F07JEF (DPTTRS).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the Hermitian positive definite tridiagonal matrix
 $A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0$
and
 $B = 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i .$

### 9.1  Program Text

Program Text (f07jsfe.f90)

### 9.2  Program Data

Program Data (f07jsfe.d)

### 9.3  Program Results

Program Results (f07jsfe.r)