MATLAB Function Reference

svd - Singular value decomposition

Syntax

s = svd(X)
[U,S,V] = svd(X)
[U,S,V] = svd(X,0)
[U,S,V] = svd(X,'econ')

Description

The svd command computes the matrix singular value decomposition.

s = svd(X) returns a vector of singular values.

[U,S,V] = svd(X) produces a diagonal matrix S of the same dimension as X, with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that X = U*S*V'.

[U,S,V] = svd(X,0) produces the "economy size" decomposition. If X is m-by-n with m > n, then svd computes only the first n columns of U and S is n-by-n.

[U,S,V] = svd(X,'econ') also produces the "economy size" decomposition. If X is m-by-n with m >= n, it is equivalent to svd(X,0). For m < n, only the first m columns of V are computed and S is m-by-m.

Examples

For the matrix

X =
     1    2
     3    4
     5    6
     7    8

the statement

[U,S,V] = svd(X)

produces

U =
    -0.1525   -0.8226   -0.3945   -0.3800
    -0.3499   -0.4214    0.2428    0.8007
    -0.5474   -0.0201    0.6979   -0.4614
    -0.7448    0.3812   -0.5462    0.0407

S =
     14.2691         0
           0    0.6268
           0         0
           0         0

V =
    -0.6414     0.7672
    -0.7672    -0.6414

The economy size decomposition generated by

[U,S,V] = svd(X,0)

produces

U =
    -0.1525   -0.8226
    -0.3499   -0.4214
    -0.5474   -0.0201
    -0.7448    0.3812
S =
    14.2691         0
          0    0.6268
V =
    -0.6414    0.7672
    -0.7672   -0.6414

Algorithm

svd uses the LAPACK routines listed in the following table to compute the singular value decomposition.

Diagnostics

If the limit of 75 QR step iterations is exhausted while seeking a singular value, this message appears:

Solution will not converge.

References

[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J.Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide (http://www.netlib.org/lapack/lug/lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999.

surfnorm

svds