calculates largest eigenvalues and eigenvectors of matrices
d = eigs(A) d = eigs(Af, n) d = eigs(.., B) d = eigs(.., B, k) d = eigs(.., B, k, sigma) d = eigs(.., B, k, sigma, opts) [d, v] = eigs(..)
a full or sparse, real or complex, symmetric or non-symmetric square matrix
a sparse, real or complex, square matrix with same dimensions as
A
positive integer: number of eigenvalues to be computed. By default,
k=min(n-1, 6)
(A
real symetric) or
k=min(n-2, 6)
(otherwise) eigen values are computed,
with n=size(A,1).
a real scalar or a string of length 2
a structure
a function
a scalar, defined only if A
is a function
a real or complex eigenvalues vector or diagonal matrix (eigenvalues along the diagonal)
real or complex eigenvector matrix
The purpose of the eigs function is to compute the largest eigenvalues of sparse, large matrices.
solves the eigenvalue problem A * v = lambda * v
. This calling returns a vector d
containing the six largest magnitude eigenvalues.
A
is either a square matrix, which can be symmetric or non-symmetric, real or complex, full or sparse.
A
should be represented by a function Af
. In this instance, a scalar n
designating
the length of the vector argument, must be defined. It must have the following header :
function y=A(x)
This function Af
must return one of the four following expressions :
if sigma is not given or is a string other than 'SM'.
if sigma is 0 or 'SM'.
for the standard eigenvalue problem, where I is the identity matrix.
for the generalized eigenvalue problem.
returns a diagonal matrix d
containing the six largest magnitude eigenvalues on the diagonal.
v
is a n by six matrix whose columns are the six eigenvectors corresponding to the returned eigenvalues.
solves the generalized eigenvalue problem A * v = lambda * B * v
with positive, definite matrix B
.
if B
is not specified, B = []
is used.
if B
is specified, B
must be the same size as A.
returns in vector d
the k
eigenvalues.
returns in vector d
the k
eigenvalues determined by sigma
.
sigma
can be either a real or complex including 0 scalar or string.
If sigma is a string of length 2, it takes one of the following values :
'LM'
compute the k
largest in magnitude eigenvalues (by default).
'SM'
compute the k
smallest in magnitude eigenvalues (same as sigma = 0).
'LA'
compute the k
Largest Algebraic eigenvalues, only for real symmetric problems.
'SA'
compute the k
Smallest Algebraic eigenvalues, only for real symmetric problems.
'BE'
compute k
eigenvalues, half from each end of the spectrum, only for real
symmetric problems.
'LR'
compute the k
eigenvalues of Largest Real part, only for real non-symmetric or
complex problems.
'SR'
compute the k
eigenvalues of Smallest Real part, only for real non-symmetric or
complex problems.
'LI'
compute the k
eigenvalues of Largest Imaginary part, only for real non-symmetric
or complex problems.
'SI'
compute the k
eigenvalues of Smallest Imaginary part, only for real non-symmetric
or complex problems.
If the opts
structure is specified, different options can be used to compute the k
eigenvalues :
tol
required convergence tolerance. By default, tol = %eps
.
maxiter
maximum number of iterations. By default, maxiter = 300
.
ncv
number of Lanzcos basis vectors to use. For real non-symmetric problems, the ncv
value must be greater or equal than 2 * k + 1
and, by default, ncv = min(max(2 * k + 1, 20), nA)
. For real symmetric or complex problems, ncv
must be greater or equal 2 * k
and, by default, ncv = min(max(2 * k, 20), nA)
with nA = size(A, 2)
.
resid
starting vector whose contains the initial residual vector, possibly from a previous run. By default,
resid
is a random initial vector.
cholB
if chol(B)
is passed rather than B
. By default, cholB
is %f.
isreal
if Af
is given, isreal
can be defined. By default, isreal
is %t.
This argument must not be indicated if A
is a matrix.
issym
if Af
is given, issym
can be defined. By default, issym
is %f.
This argument must not be indicated if A
is a matrix.
This function is based on the ARPACK package written by R. Lehoucq, K. Maschhoff, D. Sorensen, and C. Yang.
DSAUPD and DSEUPD routines for real symmetric problems,
DNAUPD and DNEUPD routines for real non-symmetric problems.
ZNAUPD and ZNEUPD routines for complex problems.
clear opts A = diag(10*ones(10,1)); A(1:$-1,2:$) = A(1:$-1,2:$) + diag(6*ones(9,1)); A(2:$,1:$-1) = A(2:$,1:$-1) + diag(6*ones(9,1)); B = eye(10,10); k = 8; sigma = 'SM'; opts.cholB = %t; d = eigs(A) [d, v] = eigs(A) d = eigs(A, B, k, sigma) [d, v] = eigs(A, B, k, sigma) d = eigs(A, B, k, sigma, opts) [d, v] = eigs(A, B, k, sigma, opts) // With sparses AS = sparse(A); BS = sparse(B); d = eigs(AS) [d, v] = eigs(AS) d = eigs(AS, BS, k, sigma) [d, v] = eigs(AS, BS, k, sigma) d = eigs(AS, BS, k, sigma, opts) [d, v] = eigs(AS, BS, k, sigma, opts) // With function clear opts function y=fn(x) y = A * x; endfunction opts.isreal = %t; opts.issym = %t; d = eigs(fn, 10, [], k, 'LM', opts) function y=fn(x) y = A \ x; endfunction d = eigs(fn, 10, [], k, 'SM', opts) function y=fn(x) y = (A - 4 * eye(10,10)) \ x; endfunction d = eigs(fn, 10, [], k, 4, opts) | ![]() | ![]() |
clear opts A = diag(10*ones(10,1)); A(1:$-1,2:$) = A(1:$-1,2:$) + diag(6*ones(9,1)); A(2:$,1:$-1) = A(2:$,1:$-1) + diag(-6*ones(9,1)); B = eye(10,10); k = 8; sigma = 'SM'; opts.cholB = %t; d = eigs(A) [d, v] = eigs(A) d = eigs(A, B, k, sigma) [d, v] = eigs(A, B, k, sigma) d = eigs(A, B, k, sigma, opts) [d, v] = eigs(A, B, k, sigma, opts) // With sparses AS = sparse(A); BS = sparse(B); d = eigs(AS) [d, v] = eigs(AS) d = eigs(AS, BS, k, sigma) [d, v] = eigs(AS, BS, k, sigma) d = eigs(AS, BS, k, sigma, opts) [d, v] = eigs(AS, BS, k, sigma, opts) // With function clear opts function y=fn(x) y = A * x; endfunction opts.isreal = %t; opts.issym = %f; d = eigs(fn, 10, [], k, 'LM', opts) function y=fn(x) y = A \ x; endfunction d = eigs(fn, 10, [], k, 'SM', opts) function y=fn(x) y = (A - 4 * eye(10,10)) \ x; endfunction d = eigs(fn, 10, [], k, 4, opts) | ![]() | ![]() |
clear opts A = diag(10*ones(10,1) + %i * ones(10,1)); A(1:$-1,2:$) = A(1:$-1,2:$) + diag(6*ones(9,1)); A(2:$,1:$-1) = A(2:$,1:$-1) + diag(-6*ones(9,1)); B = eye(10,10); k = 8; sigma = 'LM'; opts.cholB = %t; d = eigs(A) [d, v] = eigs(A) d = eigs(A, B, k, sigma) [d, v] = eigs(A, B, k, sigma) d = eigs(A, B, k, sigma, opts) [d, v] = eigs(A, B, k, sigma, opts) // With sparses AS = sparse(A); BS = sparse(B); d = eigs(AS) [d, v] = eigs(AS) d = eigs(AS, BS, k, sigma) [d, v] = eigs(AS, BS, k, sigma) d = eigs(AS, BS, k, sigma, opts) [d, v] = eigs(AS, BS, k, sigma, opts) // With function clear opts function y=fn(x) y = A * x; endfunction opts.isreal = %f; opts.issym = %f; d = eigs(fn, 10, [], k, 'LM', opts) function y=fn(x) y = A \ x; endfunction d = eigs(fn, 10, [], k, 'SM', opts) function y=fn(x) y = (A - 4 * eye(10,10)) \ x; endfunction d = eigs(fn, 10, [], k, 4, opts) | ![]() | ![]() |
Version | Description |
5.4.0 | Function introduced. Deprecates dnaupd, dneupd, dsaupd, dseupd, znaupd and zneupd. |