svd (MATLAB Functions)

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

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.

Real
Complex

X double
DGESVD
ZGESVD

X single
SGESVD
CGESVD

	Real	Complex
`X` `double`	`DGESVD`	`ZGESVD`
`X` `single`	`SGESVD`	`CGESVD`

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