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Locally Optimal Block Preconditioned Conjugate Gradient

version 4.18 (420 KB) by Andrew Knyazev
LOBPCG solves Hermitian partial generalized eigenvalue problems using preconditioning, as well as PCA


Updated 26 Apr 2021

From GitHub

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Source : in blopex_tools/matlab/lobpcg/lobpcg.m

This main function LOBPCG is a version of the preconditioned conjugate gradient method (Algorithm 5.1) described in A. V. Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, SIAM Journal on Scientific Computing 23 (2001), no. 2, pp. 517-541.
A C-version of this code is a part of the
package and is available, e.g., in SLEPc and HYPRE. A scipy version is
Tested in MATLAB 6.5-7.13-R2019a and available Octave 3.2.3-3.4.2.

Cite As

Andrew Knyazev (2021). Locally Optimal Block Preconditioned Conjugate Gradient (, GitHub. Retrieved .

Comments and Ratings (24)

Amartya Banerjee

@Andrew Knyazev: Thank you very much -- that does seem to have fixed the issues ! This is a very useful algorithm and implementation that I have found useful time and again over the years.

Andrew Knyazev

The update 3.18 has propagated here now. The updated function is blopex_tools/matlab/lobpcg/lobpcg.m

Andrew Knyazev

@Amartya Banerjee I have updated the code at
just removing the corresponding failing check.
I have also updated the repo download here, but it does not propagate somehow, still using the previous revision 3.17. Please download the updated function from the link above. Let me know if it fixes the issue.

Amartya Banerjee

Hi Andrew, I seem to be running into the same problems as the other user (Ioannis Koutis): For the most recent version of the code, function handles are not working. I am using Matlab R2019a (Update 9). I get a message which says: The size of operatorA is not the same as the number of rows of blockVectorX. Earlier versions of the code seem to be working just fine.

Andrew Knyazev

@Lee Hively There is no separate public generic Fortran 90 version to my knowledge, but there is a built-in code in open-source ABINIT

Lee Hively

Is a FORTRAN90 version of LOBPCG available? If so, where can I get it?
Please respond to, since I seldom access the mathworks website.
Thanks, Lee Hively PhD

Andrew Knyazev

@Ioannis Koutis - thanks for using the code! Could you please give an example and specify your MATLAB version?

In MATLAB Version: (R2020a) Update 2 the following runs fine for me:
A = delsq(numgrid('S',21)); n=size(A,1);
Afun = @(x)(A*x);

Ioannis Koutis

The latest version cannot handle the case when A - the left side operator is given by a function handle. Previous versions worked correctly.

Lautaro Parada

Andrew Knyazev

% Revision 4.17 adds support for single precision, e.g.,
% A = diag(1:100); B = single(diag(101:200));
% [blockVectorX,lambda]=lobpcg(randn(100,2),A,1e-5,15,2);
% A = diag(1:100); B = diag(101:200);
% [blockVectorX,lambda]=lobpcg(randn(100,2,'single'),A,1e-5,15,2);

Andrew Knyazev

Revision 4.16 adds support for distributed or codistributed arrays available in MATLAB BigData toolbox, e.g.,:

A = codistributed(diag(1:100)); B = codistributed(diag(101:200));

Andrew Knyazev

The code is now distributed from as blopex_tools/matlab/lobpcg/lobpcg.m under MIT License and Apache License 2.0

Andrew Knyazev

The code includes the license line
License: GNU LGPL ver 2.1 or above
which I intend to change into
MIT License and Apache License 2.0
as soon as I find out how.

Andrew Knyazev

I have added a few "gather" commands to this reference MATLAB code of LOBPCG so it also now runs with distributed or codistributed matrices, copy the modified code from the attachment to
and run

if true
A = distributed(diag(1:100));

Andrew Knyazev

LOBPCG is designed to be simple for distributed computing, and multiple implementations, including GPU, are already available for many years, e.g.,:

The provided reference MATLAB code can also be used, after technical modifications, for computing singular vectors or eigenvectors in the TALL ARRAY format - the new functionality not currently provided by MATLAB or any of the toolboxes. The modifications are necessary, because of some current TALL ARRAY limitations, e.g., tall(rand(10,2))\diag([1 2]) is not supported and must be substituted with tall(rand(10,2))*inv(diag([1 2])).

Please let me know if anyone cares about this functionally in MATLAB and wants me to modify the code accordingly to support the TALL ARRAY format for the eigenvectors in my LOBPCG code.

Andrew Knyazev

LOBPCG can be easily adopted to compute partial SVD and PCA for a data matrix X without ever computing its covariance matrix X'*X, i.e. in matrix-free fashion. The main calculation in LOBPCG is evaluation of a function of the product X'*(X*v) of the covariance matrix X'*X and the block-vector v. PCA needs the largest eigenvalues of the covariance matrix, while LOBPCG is typically implemented to calculate the smallest ones. A simple work-around is to negate the function, substituting -X'*(X*R) for X'*(X*R) and thus reversing the order of the eigenvalues, since LOBPCG does not care if the matrix of the eigenvalue problem is positive definite or not.

A possibly competing alternative to LOBPCG is to try EIGS, but EIGS is not used in PCA, e.g., since its code cannot be embedded or work directly with tall arrays, while LOBPCG can do both being a pure MATLAB code.

LOBPCG also supports sparse data matrix X, in contrast to PCA. The example below demonstrates that LOBPCG starts outperforming SVD, EIG, PCA, and partial PCA even for full (non-sparse) data matrix X, for matrix sizes above 5,000, when only 1 principle component is needed.

clear all; n = 6000; m = 5000; Xs = sprandn(n,m,1e-2); X = full(Xs);
tic; [U,S] = svd(X,'econ'); ttoc=toc;
clear U S
fprintf('SDV Time %i Sec \n',ttoc);
tic; A = X'*X; [V,D] = eig(A); ttoc=toc; %faster, but may be less accurate
clear A %V D
fprintf('EIG Time %e Sec \n',ttoc);
p = 1; % Number of principle components to compute
tic; [coeff] = pca(X,'Centered',false, 'Algorithm', 'eig'); ttoc=toc;
fprintf('All PCA Time %e Sec, Error %e \n',ttoc,...
tic; [coeffp] = pca(X,'Centered',false, 'NumComponents',p); ttoc=toc;
fprintf('Partial (%i principal component) PCA Time %e Sec, Error %e \n',p,ttoc,...
funA = @(v)(-(X*v)'*X)'; % using full data-matrix X
tic; [blockVectorX,lambda]=lobpcg(randn(m,p+p),@(v)funA(v),1e-10,500); ttoc=toc;
fprintf('LOBPCG full data Time %e Sec, Error %e \n',ttoc,...
funAs = @(v)(-(Xs*v)'*Xs)'; % using sparse data-matrix Xs
tic; [blockVectorX,lambda]=lobpcg(randn(m,p+p),@(v)funAs(v),1e-10,500); ttoc=toc;
fprintf('LOBPCG sparse data Time %e Sec, Error %e \n',ttoc,...

Andrew Knyazev

To enable running the code in single-precision, edit in a few spots:

Andrew Knyazev

"I encounter some severe differences in memory requirements comparing lobpcg.m with lobpcg from hypre/ij. Is there a way to use hypre with the same requirements as lobpcg.m?"

High memory requirements in hypre/ij in these tests come from hypre BoomerAMG preconditioning, not from the LOBPCG. For my full answer please see my reply at


I encounter some severe differences in memory requirements comparing lobpcg.m with lobpcg from hypre/ij. Is there a way to use hypre with the same requirements as lobpcg.m? For full explanation see

Andrew Knyazev

"Is there any reason why LOBPCG might not work for generalized eigenvalue problems with large sparse, symmetric matrices..."

Very slow convergence is an expected normal behavior of lobpcg for such a problem, without preconditioning. For detailed explanations and possible solutions, see


Is there any reason why LOBPCG might not work for generalized eigenvalue problems with large sparse, symmetric matrices (size 70 000 x 70 000, with 4.5 million non-zero values)? It has been very efficient for smaller identical problems (reducing the size of these sparse matrices to 10 000 x 10 000), although hasn't worked when I tried it on a problem of that size. In both cases I used a random matrix as an initial guess for the eigenvectors. see

Andrew Knyazev

"One question: Is there any way to compute the lowest eigenpairs above a specific value, as in eigs one can choose a SIGMA to find eigenvalues around it?"

In eigs, the SIGMA-option actually solves the so-called "shift-and-invert" problem, see . In LOBPCG, this option is not directly supported, but can be implemented by a user, supplying the corresponding functions to LOBPCG.


Nice work, really fast and very efficient. Since eigs crashes my cluster because of memory requirements, this seems to be a much better choice. One question:
Is there any way to compute the lowest eigenpairs above a specific value, as in eigs one can choose a SIGMA to find eigenvalues around it?


MATLAB Release Compatibility
Created with R2019a
Compatible with R2007b and later releases
Platform Compatibility
Windows macOS Linux

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