|
|
NAG[f08jcc] NAG[nag_dstevd] - All eigenvalues and optionally all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer)
|
|
Calling Sequence
f08jcc(job, d, e, z, 'n'=n, 'fail'=fail)
nag_dstevd(. . .)
Parameters
|
|
job - String;
|
|
|
|
On entry: indicates whether eigenvectors are computed.
|
|
|
 Only eigenvalues are computed.
|
|
|
 Eigenvalues and eigenvectors are computed.
|
|
|
Constraint:   "Nag_DoNothing" or "Nag_EigVecs". .
|
|
|
|
d - Vector(1..dim, datatype=float[8]);
|
|
|
|
Note: the dimension, dim, of the array d must be at least  .
|
|
|
On entry: the  diagonal elements of the tridiagonal matrix  .
|
|
|
On exit: the eigenvalues of the matrix  in ascending order.
|
|
|
|
e - Vector(1..dim, datatype=float[8]);
|
|
|
|
Note: the dimension, dim, of the array e must be at least  .
|
|
|
On exit: the array is overwritten with intermediate results.
|
|
|
|
z - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
|
|
|
|
Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
|
|
|
If  , z is not referenced.
|
|
|
|
'n'=n - integer; (optional)
|
|
|
|
Default value: the dimension of the array d.
|
|
|
On entry:  , the order of the matrix  .
|
|
|
Constraint:  . .
|
|
|
|
'fail'=fail - table; (optional)
|
|
|
|
The NAG error argument, see the documentation for NagError.
|
|
|
|
|
Description
|
|
|
|
Purpose
|
|
nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the  or  algorithm.
|
|
|
Error Indicators and Warnings
|
|
"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_BAD_PARAM"
On entry, argument  had an illegal value.
"NE_CONVERGENCE"
The algorithm failed to converge,  elements of an intermediate tridiagonal form did not converge to zero.
"NE_INT"
On entry,  . Constraint:  .
"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 consult NAG for assistance.
|
|
|
Accuracy
|
|
The computed eigenvalues and eigenvectors are exact for a nearby matrix  , where
 = O(epsilon)*LinearAlgebra[Norm](T, 2)](/support/helpjp/helpview.aspx?si=10424/file08255/math224.png)
and  is the machine precision.
If ![lambda[i]](/support/helpjp/helpview.aspx?si=10424/file08255/math231.png) is an exact eigenvalue and ![(lambda˜)[i]](/support/helpjp/helpview.aspx?si=10424/file08255/math233.png) is the corresponding computed value, then
![|(lambda˜)[i],-,lambda[i]|<=c(n)epsilon||T||[2]](/support/helpjp/helpview.aspx?si=10424/file08255/math236.png)
where  is a modestly increasing function of  .
If ![z[i]](/support/helpjp/helpview.aspx?si=10424/file08255/math245.png) is the corresponding exact eigenvector, and ![(z˜)[i]](/support/helpjp/helpview.aspx?si=10424/file08255/math247.png) is the corresponding computed eigenvector, then the angle ![theta((z˜)[i],,,z[i])](/support/helpjp/helpview.aspx?si=10424/file08255/math249.png) between them is bounded as follows:
![theta((z˜)[i],,,z[i])<=(c(n)epsilon||T||[2])/(min |lambda[i],-,lambda[j]|)](/support/helpjp/helpview.aspx?si=10424/file08255/math252.png)
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
|
|
|
Further Comments
|
|
There is no complex analogue of this function.
|
|
|
|
Examples
|
|
|
>
|
job := "Nag_EigVecs":
n := 4:
d := Vector([1, 4, 9, 16], datatype=float[8]):
e := Vector([1, 2, 3, 7.660404805384633e-312], datatype=float[8]):
z := Matrix(4, 4, datatype=float[8]):
NAG:-f08jcc(job, d, e, z, 'n' = n):
|
|
|
