nag_dsbtrd (f08hec) (PDF version)
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f08 Chapter Introduction
NAG C 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  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).

9  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.

9.1  Program Text

Program Text (f08hece.c)

9.2  Program Data

Program Data (f08hece.d)

9.3  Program Results

Program Results (f08hece.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012