Calculates singular value decompositions of matrices.
Option
PRINT = string tokens |
Printed output required (left, singular, right); default * i.e. no printing |
|---|
Parameters
INMATRIX = matrices |
Matrices to be decomposed |
|---|---|
LEFT = matrices |
Left-hand matrix of each decomposition |
SINGULAR = diagonal matrices |
Singular values (middle) matrix |
RIGHT = matrices |
Right-hand matrix of each decomposition |
Description
Suppose that we have a rectangular matrix A with m rows and n columns, and that p is the minimum of m and n. The singular value decomposition can be defined as
m An = mUp p Sp pVn′
The diagonal matrix S contains the p singular values of A, ordered such that
s1 ≥ s2 ≥ … ≥ sp ≥ 0
The matrices U and V contain the left and right singular vectors of A, and are orthonormal:
U′U = V′V = Ip
The smaller of U and V will be orthogonal. So, if A has more rows than columns, m>n, p=n and VV′=Ip.
The least-squares approximation of rank r to A can be formed as
Ar = Ur Sr Vr′
where Ur and Vr are the first r columns of U and V, and Sr contains the first r singular values of A (Eckart & Young 1936).
The INMATRIX parameter specifies the matrices to be decomposed. The algorithm uses Householder transformations to reduce A to bi-diagonal form, followed by a QR algorithm to find the singular values of the bi-diagonal matrix (Golub & Reinsch 1971). The other parameters allow you to save the component parts of the decomposition: LEFT, SINGULAR and RIGHT for U, S and V respectively.
The PRINT option allows you to print any of the components of the decomposition; by default, nothing is printed. If any of the matrices is to be printed, all p columns are shown, even if you are storing only the first r columns.
Genstat will decide how many columns and singular values r to store, and will store that number for any of the components that you specify. If none of the matrices in the LEFT, SINGULAR and RIGHT lists has been declared in advance, the full number of singular values (r=p) is stored; otherwise Genstat sets r to the maximum number of columns contained in any of the matrices. If r<p, the first r singular values will be saved, along with the corresponding columns of singular vectors.
One practical application of the singular value decomposition is to form generalized inverses of matrices. If you use the singular value decomposition you obtain the Moore-Penrose generalized inverse, sometimes called the pseudo-inverse, and this is the method used by the GINVERSE procedure.
Option: PRINT.
Parameters: INMATRIX, LEFT, SINGULAR, RIGHT.
References
Eckart, C. & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211-218.
Golub, G.H. & Reinsch, C. (1971). Singular value decomposition and least squares solutions. Numerische Mathematik, 14, 403-420.
See also
Directives: DIAGONALMATRIX, MATRIX, NAG, FLRV, QRD.
Functions: SVALUES, LSVECTORS, RSVECTORS.
Commands for: Calculations and manipulation,
Multivariate and cluster analysis.
Example
" Genstat example SVD-1: Singular value decomposition of matrices.
The SVD directive calculates singular value decompostitions of matrices
using Householder transformations to reduce the input matrix to
bi-diagonal form, followed by a QR algorithm to find the singular values
of the bi-diagonal matrix.
For this example, for matrix h, we wish to find u, v and s such that:
h = u s v
(16x7) (16x7) (7x7) (7x7)
We need to declare and read the values of matrix h but u,s and v will
be declared and calculated by the SVD directive.
"
MATRIX [ROWS=16; COLUMNS=7] h
READ h
16.5 24.8 106 147 1112 905 494
4.2 13.3 122 90 982 669 954
11.6 24.7 340 242 808 609 645
18.1 34.2 184 293 1668 901 602
6.9 41.5 173 191 1534 1368 780
13.0 35.7 477 220 1566 1183 788
2.5 8.8 68 103 1017 724 468
3.6 12.7 42 28 1457 1102 637
16.8 26.6 289 186 1509 787 697
10.8 43.2 255 226 1494 955 765
9.7 51.8 286 355 1902 1386 862
10.3 39.7 266 283 1056 1036 776
9.4 19.4 522 267 1674 1392 848
5.0 23.0 157 144 1530 1281 488
5.1 22.9 85 148 1206 756 483
12.5 27.6 524 217 1496 1003 739 :
PRINT h
"
Carry out the decomposition, printing the singular values
and saving the component parts into u, s and v.
Print the components one after the other.
"
SVD [PRINT=s] h; LEFT=u; SINGULAR=s; RIGHT=v
PRINT [SERIAL=yes] s,u,v
"
To save a subset of the total number of singular values, the
diagonal matrix s, must be declared explicitly. From the previous
output, it is apparent that singular values 6 and 7 may be ommitted.
To select only the first five singular values, declare ss to be a
5x5 diagonal matrix before carrying out the decomposition.
Since diagonal matrices are square matrices, Genstat needs only
the number of rows to be declared.
"
DIAGONALMATRIX [ROWS=5] ss
"
Carry out the decomposition and print the results.
"
SVD h; LEFT=us; SINGULAR=ss; RIGHT=vs
PRINT [SERIAL=y] ss,us,vs