spEigv#

Purpose#

Computes a specified number of eigenvalues and eigenvectors of a square, sparse matrix a.

Format#

{ va, ve } = spEigv(a, nev, which, tol, maxit, ncv)#
Parameters:

  • a (sparse matrix) – NxN square, sparse matrix.

  • nev (scalar) – number of eigenvalues to compute.

  • which (string) – may be one of the following: "LM" largest magnitude, "LR" largest real, "LI" largest imaginary, "SR" smallest real, or “SI” smallest imaginary. Default input 0, sets which to "LM".

  • tol (scalar) – tolerance for eigenvalues. Default input 0, sets tol to 1e-15.

  • maxit (scalar) – maximum number of iterations. Default input 0, sets maxit to \(nev * (\text{columns of a}) * 100\).

  • ncv (scalar) – size of Arnoldi factorization. The minimum setting is the greater of \(nev+2\) and 20. See Remarks on how to set ncv. Default input 0, sets ncv to \(2 * (nev+1)\).

Returns:

  • va (nevx1 dense vector) – containing the computed eigenvalues of input matrix a.

  • ve (Nxnev dense matrix) – containing the corresponding eigenvectors of input matrix a.

Examples#

// Set random seed
rndseed 3456;

// Declare sparse matrix a
sparse matrix a;

// Create random matrix x
x = 10*rndn(5, 5);

// Convert x to dense matrix
a = densetosp(x, 4);

    21.276135  5.4078872 -19.817044  9.6771132 -19.211952
    0.0000000 -4.4011007  10.445221 -5.1742289 -16.336474
a = 0.0000000 -20.853017  7.6285434  0.0000000 -15.626397
   -12.637055  8.1227002  0.0000000 -8.7817892  0.0000000
    0.0000000 -7.8181517  15.326816  0.0000000  0.0000000

/*
** This call is equivalent to calling
** { va, ve } = spEigv(a, 2,"LM", 1e-15, 2*5*100, 5);
*/
{ va, ve } = spEigv(a, 2, 0, 0, 0, 0);

va = 21.089832
    -3.4769986 + 20.141970i

ve = -0.92097057   0.29490584 - 0.38519280i
     -0.10091920  -0.18070330 - 0.38405816i
     0.061241324   0.24121182 - 0.56419722i
      0.36217049  0.017643612 + 0.26254313i
     0.081917964  -0.31466284 - 0.19936942i

Below we show that the first eigenvalue times the corresponding eigenvector (1) equals the input matrix times the first eigenvector (2).

(1) va[1]*ve[.,1]     =      (2) a*ve[.,1] =
 -19.423115                  -19.423115
 -2.1283690                  -2.1283690
  1.2915693                   1.2915693
  7.6381149                   7.6381149
  1.7276361                   1.7276361

Remarks#

The ideal setting for input ncv is problem dependent and cannot be easily predicted ahead of time. Increasing ncv will increase the amount of memory used during computation. For a large, sparse matrix, ncv should be small compared to the order of input matrix a. spEigv() is not thread-safe.

Technical Notes#

spEigv() implements functions from the ARPACK library.

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