Factorize a matrix for pseudo-inverse to solve the normal equation:
A*x = b
There are two advantages of pseudo-inverse compared to MATLAB pinv:
- PINV requires costly SVD
- PINV does not operated with sparse matrix.
The solution x minimize the 2-norm of the residual |Ax - b|.
In case of underdetermined system, i.e., rank(A) < length(x), the solution returned by pseudoinverse(A)*b is the least 2-norm among all solutions. Note that this property does *not* meet if backslash operator is used: x = A\b.
Method: Use QR factorization on both source and destination space. The factorized result is stored in object that can be used later to multiply with any target space vectors (RHS).
Inspired from FACTORIZE http://www.mathworks.com/matlabcentral/fileexchange/24119
Bruno Luong (2020). Pseudo-inverse (https://www.mathworks.com/matlabcentral/fileexchange/25453-pseudo-inverse), MATLAB Central File Exchange. Retrieved .
One big problem with pseudo-inverse; it’s a discontinuous mapping of the data when the matrix is not full rank. In other words, the pseudo-inverse of a rank deficient matrix is sensitive to noisy data. See Golub , Matrix Computation 4th edition section 5.5.5. You need regularization.
Detect SuiteSparseQR package to carry out QR factorization on sparse matrix, not available for Matlab prior 2009B (?)
New methods supported: left/right multiplication, conjugate, transpose, and complex-transpose
Change the code structure (private function instead of static methods) for better compatible with older Matlab versions
New description + Iterative method with Tkhonov regularization
Inspired: Free-knot spline approximation