# pinv¶

## Purpose¶

Computes the Moore-Penrose pseudo-inverse of a matrix, using the singular value decomposition. This pseudo-inverse is one particular type of generalized inverse.

## Format¶

y = pinv(x)
Parameters:x (NxM matrix) – data
Returns:y (MxN matrix) –

satisfies the 4 Moore-Penrose conditions:

 $$xyx = x$$ $$yxy = y$$ $$xy$$ is symmetric $$yx$$ is symmetric

## Global Input¶

_svdtol: (scalar), any singular values less than _svdtol are treated as zero in determining the rank of the input matrix. The default value for _svdtol is 1.0e-13.

## Global Output¶

_svderr: (scalar), if not all of the singular values can be computed _svderr will be nonzero.

## Examples¶

pinv() can be used to solve an undertermined least squares problem.

// Create an underdetermined system of equations 'A'
A = rndn(4, 5);

// Create a right hand side
b = rndn(4, 1);

if rank(A) < cols(A);
print "A does not have full rank, using pinv to solve";
Api = pinv(A);
x = Api*b;
else;
print "A has full rank, solve with '/' operator";
x = b/A;
endif;


Least squares problems with full rank can also be solved with the GAUSS functions: ols(), olsqr() and olsqr2().

svd.src