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svd

Fixed-point singular value decomposition

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Syntax

S = svd(A)
[U,S,V] = svd(A)
[U,S,V] = svd(A,0)
[U,S,V] = svd(A,'econ')
[___] = svd(___,sigmaForm)

Description

example

S = svd(A) returns the singular values of matrix A in descending order.

example

[U,S,V] = svd(A) performs a singular value decomposition of matrix A, such that A = U*S*V'. S is a diagonal matrix of the same dimension as A with nonnegative diagonal elements in decreasing order. U and V are unitary matrices.

example

[U,S,V] = svd(A,0) produces an economy-size decomposition of A. If A is an m-by-n matrix, then:

  • m > n — Only the first n columns of U are computed, and S is n-by-n.

  • m <= n — svd(A,0) is equivalent to svd(A).

example

[U,S,V] = svd(A,'econ') produces a different economy-size decomposition of A. If A is an m-by-n matrix, then:

  • m >= n — svd(A,'econ') is equivalent to svd(A,0).

  • m < n — Only the first m columns of V are computed, and S is m-by-m.

example

[___] = svd(___,sigmaForm) optionally specifies the output format for the singular values. You can use this option with any of the previous input or output combinations. Specify 'vector' to return the singular values as a column vector. Specify 'matrix' to return the singular values in a diagonal matrix.

Examples

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Compute the singular values of a full rank fixed-point matrix.

A = fi([1 0 1; -1 -2 0; 0 1 -1])
A = 

     1     0     1
    -1    -2     0
     0     1    -1

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 14

Compute the singular values.

s = svd(A)
s = 

    2.4606
    1.6995
    0.2391

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

The singular values are returned in a column vector in decreasing order.

Find the singular value decomposition of the rectangular fixed-point matrix A.

Define the rectangular matrix A.

m = 4;
n = 2;
A = fi(10*randn(m,n))
A = 

    5.5278    6.6016
   10.3911   -0.6787
  -11.1763   -1.9521
   12.6064   -2.1763

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 11

Find the singular value decomposition of the fixed-point matrix A.

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

    0.2885    0.8724   -0.2790    0.2789
    0.5016   -0.1879   -0.6306   -0.5617
   -0.5478   -0.1725   -0.7243    0.3816
    0.6043   -0.4169         0    0.6790

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 

   20.5887         0
         0    7.1458
         0         0
         0         0

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 

    0.9979   -0.0640
    0.0640    0.9979

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Confirm the relation A = U*S*V'.

U*S*V'
ans = 

    5.5279    6.6016
   10.3912   -0.6788
  -11.1763   -1.9521
   12.6065   -2.1764

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 99
        FractionLength: 76

Calculate the complete and economy-size decomposition of a rectangular fixed-point matrix.

Define the fixed-point matrix A.

m = 5;
n = 3;
A = fi(10*randn(m,n));
A = 

   15.3262  -10.8906    0.8594
   -7.6963    0.3252  -14.9160
    3.7139    5.5254   -7.4229
   -2.2559   11.0059  -10.6162
   11.1738   15.4424   23.5049

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 10

Compute the complete decomposition.

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

    0.1418   -0.7180    0.6097    0.2442    0.1816
   -0.4692    0.1854    0.0942   -0.0545    0.8565
   -0.0998    0.1822    0.5982   -0.7456   -0.2074
   -0.2180    0.5203    0.4894    0.6176   -0.2466
    0.8380    0.3824    0.1481         0    0.3600

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 

   34.5589         0         0
         0   22.3047         0
         0         0   14.0854
         0         0         0
         0         0         0

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 

    0.4418   -0.3881    0.8088
    0.2400    0.9199    0.3103
    0.8644   -0.0570   -0.4996

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Compute the economy-size decomposition.

[U,S,V] = fixed.svd(A,'econ')
U = 

    0.6482    0.5096    0.5333
    0.4510    0.2666   -0.7651
   -0.0660   -0.1794    0.3477
    0.5576   -0.5920    0.0698
   -0.2473    0.5353    0.0671

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 

   27.5262         0         0
         0   16.7103         0
         0         0   12.5909

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 

    0.2309   -0.6075   -0.7600
   -0.5933   -0.7070    0.3849
    0.7711   -0.3620    0.5237

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Use the expected result A = U*S*V' to determine the relative error of the calculation.

relativeError = norm(double(U*S*V'-A))/norm(double(A))
relativeError =

   1.3377e-05

Create a 3-by-3 magic square matrix and calculate the singular value decomposition. By default, the fixed.svd function returns the singular values in a diagonal matrix when you specify multiple outputs.

Define the matrix A.

m = 3; n = m;
A = fi(magic(m))
A = 

     8     1     6
     3     5     7
     4     9     2

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 11

Compute the singular value decomposition.

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

    0.5774   -0.7071   -0.4082
    0.5773    0.0000    0.8165
    0.5773    0.7071   -0.4083

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 

   15.0000         0         0
         0    6.9282         0
         0         0    3.4640

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 

    0.5774   -0.4083   -0.7071
    0.5773    0.8165   -0.0000
    0.5773   -0.4082    0.7071

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Specify the 'vector' option to return the singular values in a column vector.

[U,S,V] = svd(A,'vector')
U = 

    0.5774   -0.7071   -0.4082
    0.5773    0.0000    0.8165
    0.5773    0.7071   -0.4083

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 

   15.0000
    6.9282
    3.4640

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 

    0.5774   -0.4083   -0.7071
    0.5773    0.8165   -0.0000
    0.5773   -0.4082    0.7071

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

If you specify one output argument, such as S = svd(A), then svd switches behavior to return the singular values in a column vector by default. In that case, you can specify the 'matrix' option to return the singular values as a diagonal matrix.

Input Arguments

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Input matrix, specified as a matrix. A can be a fixed-point or scaled double fi data type.

Data Types: fi
Complex Number Support: Yes

Output format of singular values, specified as one of these values:

  • 'vector' — S is a column vector. This behavior is the default when you specify one output, S = svd(A).

  • 'matrix' — S is a diagonal matrix. This behavior is the default when you specify multiple outputs, [U,S,V] = svd(A).

Example: [U,S,V] = svd(X,'vector') returns S as a column vector instead of a diagonal matrix.

Example: S = svd(X,'matrix') returns S as a diagonal matrix instead of a column vector.

Data Types: char | string

Output Arguments

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Left singular vectors, returned as the columns of a matrix.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

Singular values, returned as a diagonal matrix or column vector. The singular values are nonnegative and returned in decreasing order.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

Right singular vectors, returned as the columns of a matrix.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

Tips

To have full control over the fixed-point types, use the fixed.svd function.

Algorithms

The svd function adjusts the data type of a fixed-point input to avoid overflow and increase precision. The fraction length of the singular vectors S is adjusted to a minimum of 16, and the word length is increased to avoid overflow with a minimum of 32. The word length of the left and right singular vectors U and V are the same as the word length of S. The fraction length of U and V is two less than the word length.

Extended Capabilities

Version History

Introduced in R2022b

See Also

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