# 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]);