# gi_vens

```CODE F2:
Gi_vens() - SHORT WRITE-UP
NAME:
Gi_vens() - Givens method for the calculation of a single
eigenvalue and the related eigenvector for a
symetric tridiagonal matrix.
SYNOPSIS:
#include
#include
#include
#include
&lt;stdio.h&gt;
&lt;math.h&gt;
&lt;alloc.h&gt;
&lt;stdlib.h&gt; /* For ANSI compability. */
double Gi_vens(double *a, double *b, int n, double lower,
double upper, double *lamda, double *x,
int *flag)
DESCRIPTION:
This function computes a single eigenvalue and the related
eigen vector for the symetric tridiagonal matrix
_
_
| a
,b , 0, ..............................., 0|
| 0
0
|
| b
,a , b , 0,..............................0|
| 0
1
1
|
| 0 ,b ,a , b , 0, ..........................0|
|
1 2
2
|
| ..............................................|
|
|
|
|
| 0 ,..............................0,b
,a
|
|
n-2 n-1 |
using Givens' method. To compute the root of the characteristic equation Newton's method is used. The calculation is
performed until limit accuracy or MAXIT (= 40) iterations
are performed which one the first. The non zero elements in
the matrix is stored in two arrays, the a-array with the elements a ,.........a
and the b-array with the elements
0
n-1
b ,.........,b
.
0
n-2
REFERENCES: L W Johnson and R D Riess, Numerical Analysis, Addison
-Wesley Publishing Company, 1982, pp 123-128.
PARAMETERS:
*a
- Array (vector) storing the matrix elements.
a
i
= *(a + i),
i = 0(1)n-1.
Space for the a - vector has to be allocated in
the calling function, use
a = (double *)malloc(n*sizeof(double))
*b
- Array (vector) storing the matrix elements.
b
i
= *(b + i),
i = 0(1)n-2.
Space for the a - vector has to be allocated in
the calling function, use
b = (double *)malloc((n-1)*sizeof(double))
n
lower
upper
*lamda
*x
- Order(dimension) of system
- Lower interval boundary for eigenvalue.
- Upper interval boundary for eigenvalue.
- On input see below, for value of flag parameter,
for computing the value of the characteristic
polynomial.
On output see below, for computing the eigenvalue
and the related eigenvector.
- Array storing computed eigenvector related to the
eigenvalue.
x = *(x+i),
i
i = 0(1)n-1.
*flag - Input value of flag parameter.
= -1, the value of the characteristic polynomial
is computed for a given value of lamda and
returned through function name Gi_vens.
The elements in the Sturm sequence with indices 1 through n are returned in the xarray.
= 0, the eigenvalue is computed. It is assumed
that the eigenvalue is in the range
[lower, upper].
Returned value of flag parameter.
= -1, eigenvalue not in the interval [lower, upper].
= 0, limit value not obtained with MAXIT iterations.
= 1, the &quot;exact&quot; root is computed.
= 2, limit acuracy obtained.
The function is to be called by
value = Gi_vens(a, b, n, lower, upper,
&amp;lamda, x, &amp;flag);
EXTERNAL PARAMETERS: None.