NAG CL Interfacef08gfc (dopgtr)

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1Purpose

f08gfc generates the real orthogonal matrix $Q$, which was determined by f08gec when reducing a symmetric matrix to tridiagonal form.

2Specification

 #include
 void f08gfc (Nag_OrderType order, Nag_UploType uplo, Integer n, const double ap[], const double tau[], double q[], Integer pdq, NagError *fail)
The function may be called by the names: f08gfc, nag_lapackeig_dopgtr or nag_dopgtr.

3Description

f08gfc is intended to be used after a call to f08gec, which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. f08gec represents the orthogonal matrix $Q$ as a product of $n-1$ elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix.

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: $\mathbf{uplo}$Nag_UploType Input
On entry: this must be the same argument uplo as supplied to f08gec.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{ap}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08gec.
5: $\mathbf{tau}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: further details of the elementary reflectors, as returned by f08gec.
6: $\mathbf{q}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $n×n$ orthogonal matrix $Q$.
7: $\mathbf{pdq}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{fail}$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.
NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}>0$.
NE_INT_2
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $‖E‖2 = O(ε) ,$
where $\epsilon$ is the machine precision.

8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08gfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gfc 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.

9Further Comments

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
The complex analogue of this function is f08gtc.

10Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) ,$
using packed storage. Here $A$ is symmetric and must first be reduced to tridiagonal form by f08gec. The program then calls f08gfc to form $Q$, and passes this matrix to f08jec which computes the eigenvalues and eigenvectors of $A$.

10.1Program Text

Program Text (f08gfce.c)

10.2Program Data

Program Data (f08gfce.d)

10.3Program Results

Program Results (f08gfce.r)