# 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]);