MATLAB Function Reference

svd

Singular value decomposition

Syntax

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

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.

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 LAPACK routines to compute the singular value decomposition.

Matrix	Routine
Real	DGESVD
Complex	ZGESVD

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