nag_dsbtrd (f08hec) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_dsbtrd (f08hec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsbtrd (f08hec) reduces a real symmetric band matrix to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dsbtrd (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double d[], double e[], double q[], Integer pdq, NagError *fail)

3  Description

nag_dsbtrd (f08hec) reduces a symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation:
T = QT A Q .
The orthogonal matrix Q is determined as a product of Givens rotation matrices, and may be formed explicitly by the function if required.
The function uses a vectorizable form of the reduction, due to Kaufman (1984).

4  References

Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

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 order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     vectNag_VectTypeInput
On entry: indicates whether Q is to be returned.
Q is returned.
Q is updated (and the array q must contain a matrix on entry).
Q is not required.
Constraint: vect=Nag_FormQ, Nag_UpdateQ or Nag_DoNotForm.
3:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored.
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     kdIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, kd, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, kd, of the matrix A.
Constraint: kd0.
6:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the n by n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in ab using the same storage format as described above.
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdab max1,kd+1 .
8:     d[n]doubleOutput
On exit: the diagonal elements of the tridiagonal matrix T.
9:     e[n-1]doubleOutput
On exit: the off-diagonal elements of the tridiagonal matrix T.
10:   q[dim]doubleInput/Output
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when vect=Nag_FormQ or Nag_UpdateQ;
  • 1 when vect=Nag_DoNotForm.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On entry: if vect=Nag_UpdateQ, q must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded symmetric-definite generalized eigenproblem); otherwise q need not be set.
On exit: if vect=Nag_FormQ or Nag_UpdateQ, the n by n matrix Q.
If vect=Nag_DoNotForm, q is not referenced.
11:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
  • if vect=Nag_FormQ or Nag_UpdateQ, pdq max1,n ;
  • if vect=Nag_DoNotForm, pdq1.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, vect=value, pdq=value and n=value.
Constraint: if vect=Nag_FormQ or Nag_UpdateQ, pdq max1,n ;
if vect=Nag_DoNotForm, pdq1.
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdab=value and kd=value.
Constraint: pdab max1,kd+1 .
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 tridiagonal matrix T is exactly similar to a nearby matrix A+E, where
E2 c n ε A2 ,
cn is a modestly increasing function of n, and ε is the machine precision.
The elements of T 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 eigenvalues and eigenvectors.
The computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision.

8  Parallelism and Performance

nag_dsbtrd (f08hec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsbtrd (f08hec) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 6n2k if vect=Nag_DoNotForm with 3n3k-1/k additional operations if vect=Nag_FormQ.
The complex analogue of this function is nag_zhbtrd (f08hsc).

10  Example

This example computes all the eigenvalues and eigenvectors of the matrix A, where
A = 4.99 0.04 0.22 0.00 0.04 1.05 -0.79 1.04 0.22 -0.79 -2.31 -1.30 0.00 1.04 -1.30 -0.43 .
Here A is symmetric and is treated as a band matrix. The program first calls nag_dsbtrd (f08hec) to reduce A to tridiagonal form T, and to form the orthogonal matrix Q; the results are then passed to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of A.

10.1  Program Text

Program Text (f08hece.c)

10.2  Program Data

Program Data (f08hece.d)

10.3  Program Results

Program Results (f08hece.r)

nag_dsbtrd (f08hec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014