NAG Library Function Document

1Purpose

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

2Specification

 #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)

3Description

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 9).

4References

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

5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{a}\left[\mathit{dim}\right]$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:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{d}\left[\mathit{dim}\right]$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:    $\mathbf{e}\left[\mathit{dim}\right]$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:    $\mathbf{tauq}\left[\mathit{dim}\right]$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 unitary matrix $Q$.
9:    $\mathbf{taup}\left[\mathit{dim}\right]$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 unitary matrix $P$.
10:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

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.

8Parallelism and Performance

nag_zgebrd (f08ksc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgebrd (f08ksc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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).

10Example

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

10.1Program Text

Program Text (f08ksce.c)

10.2Program Data

Program Data (f08ksce.d)

10.3Program Results

Program Results (f08ksce.r)

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