void test01 ( void )
/******************************************************************************/
/*
Purpose:
TEST01 uses POWER_METHOD on the Fibonacci2 matrix.
Discussion:
This matrix, despite having a single dominant eigenvalue, will generally
converge only very slowly under the power method. This has to do with
the fact that the matrix has only 3 eigenvectors.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
20 July 2008
Author:
John Burkardt
*/
{
double *a;
double cos_x1x2;
double ctime;
double ctime1;
double ctime2;
int i;
int it_max;
int it_num;
double lambda;
int n = 50;
double norm;
double phi;
int seed;
double sin_x1x2;
double tol;
double *x;
double *x2;
a = fibonacci2 ( n );
seed = 123456789;
x = r8vec_uniform_01 ( n, &seed );
it_max = 300;
tol = 0.000001;
phi = ( 1.0 + sqrt ( 5.0 ) ) / 2.0;
printf ( "\n" );
printf ( "TEST01\n" );
printf ( " Use the power method on the Fibonacci2 matrix.\n" );
printf ( "\n" );
printf ( " Matrix order N = %d\n", n );
printf ( " Maximum iterations = %d\n", it_max );
printf ( " Error tolerance = %e\n", tol );
ctime1 = cpu_time ( );
power_method ( n, a, x, it_max, tol, &lambda, &it_num );
ctime2 = cpu_time ( );
ctime = ctime2 - ctime1;
printf ( "\n" );
printf ( " Number of iterations = %d\n", it_num );
printf ( " CPU time = %f\n", ctime );
printf ( " Estimated eigenvalue = %24.16f\n", lambda );
printf ( " Correct value = %24.16f\n", phi );
printf ( " Error = %e\n", r8_abs ( lambda - phi ) );
/*
X2 is the exact eigenvector.
*/
x2 = ( double * ) malloc ( n * sizeof ( double ) );
x2[0] = 1.0;
for ( i = 1; i < n; i++ )
{
x2[i] = phi * x2[i-1];
}
norm = r8vec_norm_l2 ( n, x2 );
for ( i = 0; i < n; i++ )
{
x2[i] = x2[i] / norm;
}
/*
The sine of the angle between X and X2 is a measure of error.
*/
cos_x1x2 = r8vec_dot ( n, x, x2 );
sin_x1x2 = sqrt ( ( 1.0 - cos_x1x2 ) * ( 1.0 + cos_x1x2 ) );
printf ( "\n" );
printf ( " Sine of angle between true and estimated vectors = %e\n", sin_x1x2 );
free ( a );
free ( x );
free ( x2 );
return;
}
double *r8vec_house_column ( int n, double a[], int k )
/******************************************************************************/
/*
Purpose:
R8VEC_HOUSE_COLUMN defines a Householder premultiplier that "packs" a column.
Discussion:
An R8VEC is a vector of R8's.
The routine returns a vector V that defines a Householder
premultiplier matrix H(V) that zeros out the subdiagonal entries of
column K of the matrix A.
H(V) = I - 2 * v * v'
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
21 August 2010
Author:
John Burkardt
Parameters:
Input, int N, the order of the matrix A.
Input, double A[N], column K of the matrix A.
Input, int K, the column of the matrix to be modified.
Output, double R8VEC_HOUSE_COLUMN[N], a vector of unit L2 norm which
defines an orthogonal Householder premultiplier matrix H with the property
that the K-th column of H*A is zero below the diagonal.
*/
{
int i;
double s;
double *v;
v = r8vec_zero_new ( n );
if ( k < 1 || n <= k )
{
return v;
}
s = r8vec_norm_l2 ( n+1-k, a+k-1 );
if ( s == 0.0 )
{
return v;
}
v[k-1] = a[k-1] + r8_abs ( s ) * r8_sign ( a[k-1] );
r8vec_copy ( n-k, a+k, v+k );
s = r8vec_norm_l2 ( n-k+1, v+k-1 );
for ( i = k-1; i < n; i++ )
{
v[i] = v[i] / s;
}
return v;
}