(Pseudo)-Inverse of N by N matrix with zero determinant
c++, eigen, lapack, matrix, slam-algorithm
Solution
If all you want is to solve problem of the form Ax=B (or equivalently compute products of the form A^-1 * b), then I recommend you not to compute the inverse or pseudo-inverse of A, but directly solve for Ax=b using an appropriate rank-revealing solver. For instance, using Eigen:
x = A.colPivHouseholderQr().solve(b);
x = A.jacobiSvd(ComputeThinU|ComputeThinV).solve(b);
Problem
I would like to take the inverse of a nxn matrix to use in my GraphSlam. The issues that I encountered: - `.inverse()` Eigen-library (3.1.2) doesn't allow zero values, returns `NaN` - The LAPACK (3.4.2) library doesn't allow to use a zero determinant, but allows zero values (used example code from Computing the inverse of a matrix using lapack in C) - Seldon library (5.1.2) wouldn't compile for some reason Did anyone successfully implemented an n x n matrix inversion code that allows negative, zero-values and a determinant of zero? Any good library (C++) recommendations? I try to calculate the omega in the following for GraphSlam: http://www.acastano.com/others/udacity/cs_373_autonomous_car.html Simple example: ``` [ 1 -1 0 0 ] [ -1 2 -1 0 ] [ 0 -1 1 0 ] [ 0 0 0 0 ] ``` Real example would be 170x170 and contain 0's, negative values, bigger positive values. Given simple example is used to debug the code. I can calculate this in matlab (Moore-Penrose pseudoinverse) but for some reason I'm not able to program this in C++. ``` A = [1 -1 0 0; -1 2 -1 0; 0 -1 1 0; 0 0 0 0] B = pinv(A) B= [0.56 -0.12 -0.44 0] [-0.12 0.22 -0.11 0] [-0.44 -0.11 0.56 0] [0 0 0 0] ``` For my application I can (temporarily) remove the dimension with zero's. So I am going to remove the 4th column and the 4th row. I can also do that for my 170x170 matrix, the 4x4 was just an example. A: ``` [ 1 -1 0 ] [ -1 2 -1 ] [ 0 -1 1 ] ``` So removing the 4th column and the 4th row wouldn’t bring a zero determinant. But I can still have a zero determinant if my matrix is as above. This when the sum of each row or each column is zero. (Which I will have all the time in GraphSlam) The LAPACK-solution (Moore-Penrose Inverse based) worked if the determinant was not zero (used example code from Computing the inverse of a matrix using lapack in C). But failed as a "pseudoinverse" with a determinant of zero. SOLUTION: (all credits to Frank Reininghaus), using SVD(singular value decomposition) http://sourceware.org/ml/gsl-discuss/2008-q2/msg00013.html Works with: - Zero values (even full 0 rows and full 0 columns) - Negative values - Determinant of zero A^-1: ``` [0.56 -0.12 -0.44] [-0.12 0.22 -0.11] [-0.44 -0.11 0.56] ```