This example shows how to solve the system of equations ${(\mathit{A}}^{\prime }\mathit{A})\mathit{x}=\mathit{B}$ using forward and backward substitution.

Specify the input variables, A and B.

rng default;
A = gallery('randsvd', [5,3], 1000);
b = [1; 1; 1; 1; 1];

Compute the upper-triangular factor, R, of A, where $\mathit{A}=\mathit{QR}$.

R = fixed.qlessQR(A);

Use forward and backward substitution to compute the value of X.

X = fixed.forwardSubstitute(R,b);
X(:) = fixed.backwardSubstitute(R,X)

X = 5×1
105 ×

   -0.9088
    2.7123
   -0.8958
         0
         0

This solution is equivalent to using the fixed.qlessQRMatrixSolve function.

x = fixed.qlessQRMatrixSolve(A,b) 

x = 5×1
105 ×

   -0.9088
    2.7123
   -0.8958
         0
         0

Using a forgetting factor with the fixed.qlessQR function is roughly equivalent to the Complex- and Real Partial-Systolic Q-less QR with Forgetting Factor blocks. These blocks process one row of the input matrix at a time and apply the forgetting factor after each row is processed. The fixed.qlessQR function takes in all rows of A at once, but carries out the computation in the same way as the blocks. The forgetting factor is applied after each row is processed.

Specifying a forgetting factor is useful when you want to stream an indefinite number of rows continuously, such as reading values from a sensor array continuously, without accumulating the data without bound.

Without using a forgetting factor, the accumulation is the square root of the number of rows, so 10000 rows would accumulate to $\sqrt{10000}=100$.

A = ones(10000,3);
R = fixed.qlessQR(A)

R = 3×3

  100.0000  100.0000  100.0000
         0    0.0000    0.0000
         0         0    0.0000

To accrue with the effective height of m=16 rows, set the forgetting factor to the following.

m=16;
forgettingFactor = exp(-1/(2*m))

forgettingFactor = 0.9692

Using the forgetting factor, fixed.qlessQR would accumulate to about square root of 16.

R = fixed.qlessQR(A,forgettingFactor)

R = 3×3

    3.9377    3.9377    3.9377
         0    0.0000    0.0000
         0         0    0.0000