f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zgebrd (f08ksc)

## 1  Purpose

nag_zgebrd (f08ksc) reduces a complex $m$ by $n$ matrix to bidiagonal form.

## 2  Specification

 #include #include
 void nag_zgebrd (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, double d[], double e[], Complex tauq[], Complex taup[], NagError *fail)

## 3  Description

nag_zgebrd (f08ksc) reduces a complex $m$ by $n$ matrix $A$ to real bidiagonal form $B$ by a unitary transformation: $A=QB{P}^{\mathrm{H}}$, where $Q$ and ${P}^{\mathrm{H}}$ are unitary matrices of order $m$ and $n$ respectively.
If $m\ge n$, the reduction is given by:
 $A =Q B1 0 PH = Q1 B1 PH ,$
where ${B}_{1}$ is a real $n$ by $n$ upper bidiagonal matrix and ${Q}_{1}$ consists of the first $n$ columns of $Q$.
If $m, the reduction is given by
 $A =Q B1 0 PH = Q B1 P1H ,$
where ${B}_{1}$ is a real $m$ by $m$ lower bidiagonal matrix and ${P}_{1}^{\mathrm{H}}$ consists of the first $m$ rows of ${P}^{\mathrm{H}}$.
The unitary matrices $Q$ and $P$ are not formed explicitly but are represented as products of elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with $Q$ and $P$ in this representation (see Section 8).

## 4  References

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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     nIntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ matrix $A$.
On exit: if $m\ge n$, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix $B$, elements below the diagonal are overwritten by details of the unitary matrix $Q$ and elements above the first superdiagonal are overwritten by details of the unitary matrix $P$.
If $m, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix $B$, elements below the first subdiagonal are overwritten by details of the unitary matrix $Q$ and elements above the diagonal are overwritten by details of the unitary matrix $P$.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     d[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the diagonal elements of the bidiagonal matrix $B$.
7:     e[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)-1\right)$.
On exit: the off-diagonal elements of the bidiagonal matrix $B$.
8:     tauq[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array tauq must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the matrix $Q$.
9:     taup[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array taup must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the matrix $P$.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

The computed bidiagonal form $B$ satisfies $QB{P}^{\mathrm{H}}=A+E$, where
 $E2 ≤ c n ε A2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ is the machine precision.
The elements of $B$ themselves may be sensitive to small perturbations in $A$ or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.

The total number of real floating point operations is approximately $16{n}^{2}\left(3m-n\right)/3$ if $m\ge n$ or $16{m}^{2}\left(3n-m\right)/3$ if $m.
If $m\gg n$, it can be more efficient to first call nag_zgeqrf (f08asc) to perform a $QR$ factorization of $A$, and then to call nag_zgebrd (f08ksc) to reduce the factor $R$ to bidiagonal form. This requires approximately $8{n}^{2}\left(m+n\right)$ floating point operations.
If $m\ll n$, it can be more efficient to first call nag_zgelqf (f08avc) to perform an $LQ$ factorization of $A$, and then to call nag_zgebrd (f08ksc) to reduce the factor $L$ to bidiagonal form. This requires approximately $8{m}^{2}\left(m+n\right)$ operations.
To form the unitary matrices ${P}^{\mathrm{H}}$ and/or $Q$ nag_zgebrd (f08ksc) may be followed by calls to nag_zungbr (f08ktc):
to form the $m$ by $m$ unitary matrix $Q$
```nag_zungbr(order,Nag_FormQ,m,m,n,&a,pda,tauq,&fail)
```
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_zgebrd (f08ksc);
to form the $n$ by $n$ unitary matrix ${P}^{\mathrm{H}}$
```nag_zungbr(order,Nag_FormP,n,n,m,&a,pda,taup,&fail)
```
but note that the first dimension of the array a, specified by the argument pda, must be at least n, which may be larger than was required by nag_zgebrd (f08ksc).
To apply $Q$ or $P$ to a complex rectangular matrix $C$, nag_zgebrd (f08ksc) may be followed by a call to nag_zunmbr (f08kuc).
The real analogue of this function is nag_dgebrd (f08kec).

## 9  Example

This example reduces the matrix $A$ to bidiagonal form, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$

### 9.1  Program Text

Program Text (f08ksce.c)

### 9.2  Program Data

Program Data (f08ksce.d)

### 9.3  Program Results

Program Results (f08ksce.r)