# NAG CL Interfacef08kec (dgebrd)

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## 1Purpose

f08kec reduces a real $m×n$ matrix to bidiagonal form.

## 2Specification

 #include
 void f08kec (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double d[], double e[], double tauq[], double taup[], NagError *fail)
The function may be called by the names: f08kec, nag_lapackeig_dgebrd or nag_dgebrd.

## 3Description

f08kec reduces a real $m×n$ matrix $A$ to bidiagonal form $B$ by an orthogonal transformation: $A=QB{P}^{\mathrm{T}}$, where $Q$ and ${P}^{\mathrm{T}}$ are orthogonal matrices of order $m$ and $n$ respectively.
If $m\ge n$, the reduction is given by:
 $A =Q ( B1 0 ) PT = Q1 B1 PT ,$
where ${B}_{1}$ is an $n×n$ upper bidiagonal matrix and ${Q}_{1}$ consists of the first $n$ columns of $Q$.
If $m, the reduction is given by
 $A =Q ( B1 0 ) PT = Q B1 P1T ,$
where ${B}_{1}$ is an $m×m$ lower bidiagonal matrix and ${P}_{1}^{\mathrm{T}}$ consists of the first $m$ rows of ${P}^{\mathrm{T}}$.
The orthogonal 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_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left[\mathit{dim}\right]$double Input/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 $m×n$ 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 orthogonal matrix $Q$ and elements above the first superdiagonal are overwritten by details of the orthogonal 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 orthogonal matrix $Q$ and elements above the diagonal are overwritten by details of the orthogonal matrix $P$.
5: $\mathbf{pda}$Integer Input
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: $\mathbf{d}\left[\mathit{dim}\right]$double Output
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: $\mathbf{e}\left[\mathit{dim}\right]$double Output
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: $\mathbf{tauq}\left[\mathit{dim}\right]$double Output
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 orthogonal matrix $Q$.
9: $\mathbf{taup}\left[\mathit{dim}\right]$double Output
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 orthogonal matrix $P$.
10: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The computed bidiagonal form $B$ satisfies $QB{P}^{\mathrm{T}}=A+E$, where
 $‖E‖2 ≤ c (n) ε ‖A‖2 ,$
$c\left(n\right)$ 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

f08kec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kec 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 floating-point operations is approximately $\frac{4}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{4}{3}{m}^{2}\left(3n-m\right)$ if $m.
If $m\gg n$, it can be more efficient to first call f08aec to perform a $QR$ factorization of $A$, and then to call f08kec to reduce the factor $R$ to bidiagonal form. This requires approximately $2{n}^{2}\left(m+n\right)$ floating-point operations.
If $m\ll n$, it can be more efficient to first call f08ahc to perform an $LQ$ factorization of $A$, and then to call f08kec to reduce the factor $L$ to bidiagonal form. This requires approximately $2{m}^{2}\left(m+n\right)$ operations.
To form the $m×m$ orthogonal matrix $Q$ f08kec may be followed by a call to f08kfc . For example
`nag_lapackeig_dorgbr(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 f08kec.
To form the $n×n$ orthogonal matrix ${P}^{\mathrm{T}}$ another call to f08kfc may be made . For example
`nag_lapackeig_dorgbr(order,Nag_FormP,n,n,m,&a,pda,taup,&fail)`
but note that the first dimension of the array a, must be at least n, which may be larger than was required by f08kec.
To apply $Q$ or $P$ to a real rectangular matrix $C$, f08kec may be followed by a call to f08kgc.
The complex analogue of this function is f08ksc.

## 10Example

This example reduces the matrix $A$ to bidiagonal form, where
 $A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) .$

### 10.1Program Text

Program Text (f08kece.c)

### 10.2Program Data

Program Data (f08kece.d)

### 10.3Program Results

Program Results (f08kece.r)