svdusv

Purpose

Computes the singular value decomposition of x so that: \(x = u * s * v'\).

Format

{ u, s, v } = svdusv(x)
Parameters:

x (NxP matrix or K-dimensional array) – data whose singular values are to be computed, where the last two dimensions are NxP.

Returns:
  • u (NxN matrix or K-dimensional array) – where the last two dimensions are \(NxN\), the left singular vectors of x.
  • s (NxP diagonal matrix or K-dimensional array) – where the last two dimensions describe \(NxP\) diagonal arrays, the singular values of x arranged in descending order on the principal diagonal.
  • v (PxP matrix or K-dimensional array) – where the last two dimensions are \(PxP\), the right singular vectors of x.

Examples

// Create 6x3 matrix
x = { -9.35    15.67   -41.75,
     -13.55    40.97    15.55,
      -0.95   -17.03    40.15,
       8.15    -9.73    13.15,
       2.35   -36.73   -43.55,
      13.35     6.87    16.45  };

// Perform matrix decomposition
{ u, s, v } = svdusv(x);

After the code above, the outputs will have the following values;

u =  0.44    -0.49    -0.06     0.36    -0.24     0.61
    -0.35    -0.60    -0.28     0.12     0.65    -0.08
    -0.41     0.46    -0.53     0.07     0.03     0.58
    -0.12     0.25     0.24     0.91     0.08    -0.18
     0.67     0.35    -0.13    -0.02     0.64     0.05
    -0.23     0.04     0.75    -0.17     0.33     0.50

s = 79.03     0.00     0.00
     0.00    60.19     0.00
     0.00     0.00    17.16
     0.00     0.00     0.00
     0.00     0.00     0.00
     0.00     0.00     0.00

v = -0.02     0.26     0.97
    -0.32    -0.91     0.24
    -0.95     0.31    -0.10

Remarks

  1. If x is an array, the resulting arrays u, s and v will contain their respective results for each of the corresponding 2-dimensional arrays described by the two trailing dimensions of x. In other words, for a 10x4x5 array x:

    • u will be a 10x4x4 array, containing the left singular vectors of each of the 10 corresponding 4x5 arrays contained in x.
    • s will be a 10x4x5 array, containing the singular values.
    • v will be a 10x5x5 array containing, the right singular vectors.
  2. Error handling is controlled by the trap command. If not all of the singular values can be computed:

    trap 0 terminate with an error message
    trap 1 set the first element of s to a scalar missing value and continue execution
    // Turn on error trapping
    trap 1;
    
    // Compute singular value decomposition
    { u, s, v } = svdusv(x);
    
    // Check for failure or success
    if scalmiss(s[1, 1]);
       // Code for failure case
    endif;
    

    Note that in the trap 1 case, if the input to svdusv() is a multi-dimensional array and the singular values for a submatrix fail to compute, only the first value of that s submatrix will be set to a missing value. For a 3 dimensional array, you could change the if, else, elseif, endif check in the above example to:

    // Check for success or failure of each submatrix
    if ismiss(s[., 1, 1]);
    

See also

Functions svd1(), svdcusv(), svds()