svdcusv

Purpose

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

Format

{ u, s, v } = svdcusv(x)

Parameters:

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

Returns:

• u (NxN or NxP matrix or K-dimensional array) – the last two dimensions are \(NxN\) or \(NxP\), the left singular vectors of x. If \(N > P\), u is \(NxP\), containing only the \(P\) left singular vectors of x.

• s (NxP or PxP diagonal matrix or K-dimensional array) – the last two dimensions describe \(NxP\) or \(PxP\) diagonal arrays, the singular values of x arranged in descending order on the principal diagonal. If \(N > P\), s is \(PxP\).

• v (PxP matrix or K-dimensional array) – the last two dimensions are \(PxP\), the right singular vectors of x.

Examples

// Create a 10x3 matrix
x = {  -0.60     3.50     0.47,
        8.40    16.50     0.27,
       11.40     6.50     0.17,
        7.40    -0.50    -2.43,
       -9.60   -10.50     0.57,
      -17.60    -5.50     0.67,
      -12.60   -14.50     0.87,
       18.40    12.50    -1.43,
      -11.60   -19.50     0.77,
        6.40    11.50     0.07 };

// Calculate the singular values
{ u, s, v } = svdcusv(x);

After the code above, u, s and v will be equal to:

u =  0.04     0.20    -0.11
     0.36     0.38    -0.14
     0.25    -0.23    -0.44
     0.10    -0.39     0.75
    -0.29    -0.04    -0.06
    -0.33     0.57     0.35
    -0.39    -0.08    -0.14
     0.44    -0.29     0.10
    -0.44    -0.37    -0.25
     0.26     0.24    -0.07

s = 49.58     0.00     0.00
     0.00    14.96     0.00
     0.00     0.00     2.24

v =  0.70    -0.70    -0.10
     0.71     0.70     0.05
    -0.04     0.10    -0.99

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 } = svdcusv(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 svdcusv() 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 check in the above example to:

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

See also

Functions svd2(), svds(), svdusv()