If is rank deficient, then the least squares problem has infinitely many solutions (Golub and Van Loan; 1989, p. 241). However, there is a unique solution which has the smallest Euclidean norm. The APPCORT subroutine computes the minimum Euclidean-norm solution of the (rank-deficient) least squares problem by applyng a complete orthogonal decomposition by Householder transformations to the vector .

The input arguments to the APPCORT subroutine are as follows:

a

is an matrix , with , which is to be decomposed into the product of the orthogonal matrix , the upper triangular matrix , and the orthogonal matrix ,

b

is a matrix, .

sing

is an optional scalar that specifies a singularity criterion.

The APPCORT subroutine returns the following values:

prqb

is an matrix product

which is the minimum Euclidean-norm solution of the rank-deficient least squares problem .

lindep

is the number of linearly dependent columns in the matrix that are detected by applying the Householder transformations. That is, lindep is , where is the numerical rank of .

See the section COMPORT Call for information about complete orthogonal decomposition.

The following example uses the APPCORT call to solve a rank-deficient least squares problem:

/* compute solution for rank-deficient least squares problem:
      min |Ax-b|^2
   The range of A is a line; b is a point not on the line. */
A = {1  2,
     2  4,
    -1 -2};
b = {1, 3, -2};
call appcort(x,lindep,A,b);
print x;

The argument b can also be a matrix. If b is an identity matrix, then you can use the APPCORT subroutine to form a generalized inverse, as shown in the following example:

/* A has only four linearly independent columns */
A = {1 0 1 0 0,
     1 0 0 1 0,
     1 0 0 0 1,
     0 1 1 0 0,
     0 1 0 1 0,
     0 1 0 0 1 };

/* compute Moore-Penrose generalized inverse */
b = i(nrow(A));                   /* identity matrix */
call appcort(Ainv, lindep, A, b);
print Ainv;

/* verify generalized inverse conditions (Golub & Van Loan, p. 243) */
eps = 1e-12;
if any(A*Ainv*A-A > eps) |
   any(Ainv*A*Ainv-Ainv > eps) |
   any((A*Ainv)`-A*Ainv > eps) |
   any((Ainv*A)`-Ainv*A > eps) then
   msg = "Pseudoinverse conditions not satisfied";
else
   msg = "Pseudoinverse conditions satisfied";
print msg;