lapeighvb

Purpose

Computes eigenvalues and eigenvectors of a real symmetric or complex Hermitian matrix selected by bounds.

Format

{ ve, va } = lapeighvb(x, vl, vu, abstol)
Parameters:
  • x (NxN matrix) – real symmetric or complex Hermitian.
  • vl (scalar) – lower bound of the interval to be searched for eigenvalues.
  • vu (scalar) – upper bound of the interval to be searched for eigenvalues; vu must be greater than vl.
  • abstol (scalar) – the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval \([a, b]\) of width less than or equal to \(abstol + EPS*max(|a|, |b|)\), where EPS is machine precision. If abstol is less than or equal to zero, then \(EPS*||T||\) will be used in its place, where T is the tridiagonal matrix obtained by reducing the input matrix to tridiagonal form.
Returns:
  • ve (Mx1 vector) – eigenvalues, where M is the number of eigenvalues on the half open interval \([vl, vu]\). If no eigenvalues are found then s is a scalar missing value.
  • va (NxM matrix) – eigenvectors.

Examples

// Assign x
x = { 5 2 1,
      2 6 2,
      1 2 9 };

// Set lower bound to be searched
vl = 5;

// Set upper bound to be searched
vu = 10;

// Find eigenvalues within the bounds of
// vl and vu and the corresponding
// eigenvectors
{ ve, va } = lapeighvb(x, vl, vu, 0);

print "Eigenvalues = " ve;
print "Eigenvectors = " va;
Eigenvalues =   6.0000
Eigenvectors =
 -0.5774
 -0.5774
  0.5774

If you increase the value of vu to 12.

// Set lower bound to be searched
vl = 5;

// Set upper bound to be searched
vu = 12;

// Find eigenvalues within the bounds of
// vl and vu and the corresponding
// eigenvectors
{ ve, va } = lapeighvb(x, vl, vu, 0);

print "Eigenvalues = " ve;
print "Eigenvectors = " va;
Eigenvalues
  6.0000
 10.6056
Eigenvectors =
 -0.5774   0.3197
 -0.5774   0.4908
  0.5774   0.8105

Remarks

lapeighvb() computes eigenvalues and eigenvectors which are found on the half open interval \([vl, vu]\). lapeighvb() is based on the LAPACK drivers DSYEVX and ZHEEVX. Further documentation of these functions may be found in the LAPACK User’s Guide.