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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhptrd (f08gs)

Purpose

nag_lapack_zhptrd (f08gs) reduces a complex Hermitian matrix to tridiagonal form, using packed storage.

Syntax

[ap, d, e, tau, info] = f08gs(uplo, n, ap)
[ap, d, e, tau, info] = nag_lapack_zhptrd(uplo, n, ap)

Description

nag_lapack_zhptrd (f08gs) reduces a complex Hermitian matrix A$A$, held in packed storage, to real symmetric tridiagonal form T$T$ by a unitary similarity transformation: A = QTQH$A=QT{Q}^{\mathrm{H}}$.
The matrix Q$Q$ is not formed explicitly but is represented as a product of n1$n-1$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q$Q$ in this representation (see Section [Further Comments]).

References

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A$A$ is stored.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The upper or lower triangle of the n$n$ by n$n$ Hermitian matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.

None.

None.

Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
ap stores the tridiagonal matrix T$T$ and details of the unitary matrix Q$Q$.
2:     d(n) – double array
The diagonal elements of the tridiagonal matrix T$T$.
3:     e(n1${\mathbf{n}}-1$) – double array
The off-diagonal elements of the tridiagonal matrix T$T$.
4:     tau(n1${\mathbf{n}}-1$) – complex array
Further details of the unitary matrix Q$Q$.
5:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: d, 5: e, 6: tau, 7: info.

Accuracy

The computed tridiagonal matrix T$T$ is exactly similar to a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 ≤ c(n) ε ‖A‖2 , $‖E‖2≤ c(n) ε ‖A‖2 ,$
c(n)$c\left(n\right)$ is a modestly increasing function of n$n$, and ε$\epsilon$ is the machine precision.
The elements of T$T$ themselves may be sensitive to small perturbations in A$A$ or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

The total number of real floating point operations is approximately (16/3) n3 $\frac{16}{3}{n}^{3}$.
To form the unitary matrix Q$Q$ nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupgtr (f08gt):
```[q, info] = f08gt(uplo, n, ap, tau);
```
To apply Q$Q$ to an n$n$ by p$p$ complex matrix C$C$ nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupmtr (f08gu). For example,
```[ap, c, info] = f08gu('Left', uplo, 'No Transpose', ap, tau, c);
```
forms the matrix product QC$QC$.
The real analogue of this function is nag_lapack_dsptrd (f08ge).

Example

```function nag_lapack_zhptrd_example
uplo = 'L';
n = int64(4);
ap = [-2.28;
1.78 + 2.03i;
2.26 - 0.1i;
-0.12 - 2.53i;
-1.12 + 0i;
0.01 - 0.43i;
-1.07 - 0.86i;
-0.37 + 0i;
2.31 + 0.92i;
-0.73 + 0i];
[apOut, d, e, tau, info] = nag_lapack_zhptrd(uplo, n, ap)
```
```

apOut =

-2.2800 + 0.0000i
-4.3385 + 0.0000i
0.3279 - 0.1251i
-0.1413 - 0.3666i
-0.1285 + 0.0000i
-2.0226 + 0.0000i
-0.3083 + 0.1763i
-0.1666 + 0.0000i
-1.8023 + 0.0000i
-1.9249 + 0.0000i

d =

-2.2800
-0.1285
-0.1666
-1.9249

e =

-4.3385
-2.0226
-1.8023

tau =

1.4103 + 0.4679i
1.3024 + 0.7853i
1.0940 - 0.9956i

info =

0

```
```function f08gs_example
uplo = 'L';
n = int64(4);
ap = [-2.28;
1.78 + 2.03i;
2.26 - 0.1i;
-0.12 - 2.53i;
-1.12 + 0i;
0.01 - 0.43i;
-1.07 - 0.86i;
-0.37 + 0i;
2.31 + 0.92i;
-0.73 + 0i];
[apOut, d, e, tau, info] = f08gs(uplo, n, ap)
```
```

apOut =

-2.2800 + 0.0000i
-4.3385 + 0.0000i
0.3279 - 0.1251i
-0.1413 - 0.3666i
-0.1285 + 0.0000i
-2.0226 + 0.0000i
-0.3083 + 0.1763i
-0.1666 + 0.0000i
-1.8023 + 0.0000i
-1.9249 + 0.0000i

d =

-2.2800
-0.1285
-0.1666
-1.9249

e =

-4.3385
-2.0226
-1.8023

tau =

1.4103 + 0.4679i
1.3024 + 0.7853i
1.0940 - 0.9956i

info =

0

```