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## Fast and Accurate Symmetric Positive Definite Matrix Inverse Using Cholesky Decomposition

version 1.11.0.0 (2.69 KB) by
use LAPACK Cholesky to invert real positive definite symmetric matrix; faster more accurate than inv

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Updated 18 Feb 2015

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We can exploit the structure of a real, positive definite, symmetric matrix by using the Cholesky decomposition to compute the inverse. The standard MATLAB inv function uses LU decomposition which requires twice as many operations as the Cholesky decomposition and is less accurate. Rarely does one need to compute the inverse of a matrix (e.g. when solving a linear system, we should use \), but when it is needed (e.g. least squares or Kalman Filtering applications), the matrix is positive definite and symmetric. Thus, we can speed up and improve the accuracy of the computation by using the Cholesky decomposition. Because the Cholesky decomposition takes half as many operations, the algorithm will decrease run-time by about 50% for large matrices. Accuracy is also improved with some examples showing orders of magnitude improvement (see Felix Govaers's comment).

### Cite As

Eric Blake (2021). Fast and Accurate Symmetric Positive Definite Matrix Inverse Using Cholesky Decomposition (https://www.mathworks.com/matlabcentral/fileexchange/34511-fast-and-accurate-symmetric-positive-definite-matrix-inverse-using-cholesky-decomposition), MATLAB Central File Exchange. Retrieved .

### Comments and Ratings (13)

Julien Bernard

Constantin Grigo

Short addition: The method I mentioned below is only faster for small matrices ~< 400x400. For larger matrices, the invChol_mex function is faster.

Constantin Grigo

Guys, you should consider using

opts.POSDEF = true;
opts.SYM = true;
I = eye(dim_of_A);
Ainv = linsolve(A, I, opts);

Works another 20% faster than this one for me!

Luca Peluso

I have to fast-invert this matrix Z= (R+jwL) where R is a real diagonal matrix and L is a real, symmetric, positive definite matrix. Is there any way to use this kind of inversion for the Z matrix?

Thank you in advance!

Burke Rosen

I wonder if even more speed could be achieved by using intel mkl?

mmmm

works great thanks a lot! Is there any way to also get the decomposition itself from it? (this is needed in maximum likelihood applications to calculate the log determinant)

mmmm

Sanjay

Thanks for the code.

I would like to know what is the reference of this code? I mean the technical details of the mentioned fast and accurate computation. Unfortunately it is not mentioned in the code. In case someone is interested to cite it, what reference (paper/text book) should be cited?

I am curious to know the complexity analysis of this inversion method.

Felix Govaers

Thanks a lot for the code, very much appreciated. It should be noted that the Cholesky inversion is also of higher precision. This made a huge difference in my application!

I did:
err_chol_inv = A_cholinv * A - eye(matdim);
err_inv = A_inv * A - eye(matdim);

Max diff chol inv: 5.704351e-09
Max diff inv: 1.980871e-04

This can also be achieved by using Matlab code only (without the speed up advantage):
L = inv(chol(A));
A_inv = L * L';

Mikhail Konnik

Thanks for the updated code, Eric!
Works like a charm.

On a side note, and I think it is worth mentioning in the description that the speed-up is noticeable when the matrices are large (in the current example they are small). In my case, I have to solve a big least-squares problem - and invChol, literally, saves the day.

Sacha

When I run the test.m file i receive an illegal value warning in the dpotrf procedure. Any Help would be greatly appreciated

Mikhail Konnik

Excellent code, simple yet efficient, thank you!

One comment. When I compile this in Linux (x86) on MATLAB 2009, I'm getting these warnings:

/opt/matlab/bin/mex invChol_mex.c -lmwlapack

invChol_mex.c: In function ‘mexFunction’:
invChol_mex.c:75: warning: implicit declaration of function ‘dpotrf_’
invChol_mex.c:87: warning: implicit declaration of function ‘dpotri_’
invChol_mex.c:111: warning: implicit declaration of function ‘spotrf_’
invChol_mex.c:123: warning: implicit declaration of function ‘spotri_’

Not critical, of course, but still.

liu hu

##### MATLAB Release Compatibility
Created with R2011b
Compatible with any release
##### Platform Compatibility
Windows macOS Linux

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