# Documentation

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# svds

Subset of singular values and vectors

## Description

example

s = svds(A) returns a vector of the six largest singular values of matrix A.

example

s = svds(A,k) returns the k largest singular values.

example

s = svds(A,k,sigma) returns k singular values based on the value of sigma. For example, svds(A,k,'smallest') returns the k smallest singular values.

example

s = svds(A,k,sigma,opts) additionally specifies options using a structure.

example

s = svds(Afun,n,___) specifies a function handle Afun instead of a matrix A. You can optionally specify k, sigma, or opts as additional input arguments.

example

[U,S,V] = svds(___) returns the left singular vectors U, diagonal matrix S of singular values, and right singular vectors V. You can use any of the input argument combinations in previous syntaxes.

example

[U,S,V,flag] = svds(___) also returns a convergence flag. If flag is 0, then all the singular values converged.

## Examples

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Compute the six largest singular values of a sparse matrix.

A = delsq(numgrid('C',15));
s = svds(A)
s =

7.8666
7.7324
7.6531
7.5213
7.4480
7.3517

Specify a second input to compute a specific number of the largest singular values.

s = svds(A,3)
s =

7.8666
7.7324
7.6531

Compute the five smallest singular values of a sparse matrix.

A = delsq(numgrid('C',15));
s = svds(A,5,'smallest')
s =

0.5520
0.4787
0.3469
0.2676
0.1334

Create a sparse 100-by-100 Neumann matrix.

C = gallery('neumann',100);

Compute the ten smallest singular values.

ss = svds(C,10,'smallest')
ss =

0.9828
0.9049
0.5625
0.5625
0.4541
0.4506
0.2256
0.1139
0.1139
0

Compute the 10 smallest nonzero singular values. Since the matrix has a singlular value that is equal to zero, the 'smallestnz' option omits it.

snz = svds(C,10,'smallestnz')
snz =

0.9828
0.9828
0.9049
0.5625
0.5625
0.4541
0.4506
0.2256
0.1139
0.1139

Create two matrices representing the upper-right and lower-left nonzero blocks in a sparse matrix.

n = 500;
B = rand(500);
C = rand(500);

Save Afun in your current directory so that it is available for use with svds.

function y = Afun(x,tflag,B,C,n)
if strcmp(tflag,'notransp')
y = [B*x(n+1:end); C*x(1:n)];
else
y = [C'*x(n+1:end); B'*x(1:n)];
end

The function Afun uses B and C to compute either A*x or A'*x (depending on the specified flag) without actually forming the entire sparse matrix A = [zeros(n) B; C zeros(n)]. This exploits the sparsity pattern of the matrix to save memory in the computation of A*x and A'*x.

Use Afun to calculate the 10 largest singular values of A. Pass B, C, and n as additional inputs to Afun.

s = svds(@(x,tflag) Afun(x,tflag,B,C,n),[1000 1000],10)
s =

250.3248
249.9914
12.7627
12.7232
12.6988
12.6608
12.6166
12.5643
12.5419
12.4512

Directly compute the 10 largest singular values of A to compare the results.

A = [zeros(n) B; C zeros(n)];
s = svds(A,10)
s =

250.3248
249.9914
12.7627
12.7232
12.6988
12.6608
12.6166
12.5643
12.5419
12.4512

west0479 is a real-valued 479-by-479 sparse matrix. The matrix has a few large singular values, and many small singular values.

Load west0479 and store it as A.

A = west0479;

Compute the singular value decomposition of A, returning the six largest singular values and the corresponding singular vectors. Specify a fourth output argument to check convergence of the singular values.

[U,S,V,cflag] = svds(A);
cflag
cflag =

0

cflag indicates that all of the singular values converged. The singular values are on the diagonal of the output matrix S.

s = diag(S)
s =

1.0e+05 *

3.1895
3.1725
3.1695
3.1685
3.1669
0.3038

Check the results by computing the full singular value decomposition of A. Convert A to a full matrix and use svd.

[U1,S1,V1] = svd(full(A));

Plot all of the singular values of A using a logarithmic scale.

semilogy(diag(S1),'r.')
title('Singular Values of west0479')

Create a sparse diagonal matrix and calculate the six largest singular values.

A = diag(sparse([1e4*ones(1, 8) 1e4:-1:1]));
s = svds(A)
Warning: Maximum number of iterations reached. Results may be inaccurate.

s =

1.0e+04 *

1.0000
0.9999
0.9998
0.9997
0.9996
0.9995

The svds algorithm produces a warning since the maximum number of iterations were performed but the tolerance could not be met.

The most effective way to address convergence problems is to increase the maximum size of the Krylov subspace used in the calculation by using a larger value for p. Do this by passing an options structure to svds that specifies p = 50.

opts.p = 50;
s = svds(A,6,'largest',opts)
Warning: Maximum number of iterations reached. Results may be inaccurate.

s =

1.0e+04 *

1.0000
1.0000
1.0000
1.0000
1.0000
1.0000

Compute the 10 smallest singular values of a nearly singular matrix.

rng default
format shortg
B = spdiags([repelem([1; 1e-7], [198, 2]) ones(200, 1)], [0 1], 200, 200);
s1 = svds(B,10,'smallest')
Warning: Large residual norm detected. This is likely due to bad condition of
the input matrix (condition number 1.0008e+16).

s1 =

7.0945
7.0945
7.0945
7.0945
7.0945
7.0945
7.0945
7.0945
0.25927
7.0888e-16

The warning indicates that svds fails to calculate the proper singular values. The failure with svds is because of the gap between the smallest and second smallest singular values. svds(...,'smallest') needs to invert B, which leads to large numerical error.

For comparison, compute the exact singular values using svd.

s = svd(full(B));
s = s(end-9:end)
s =

0.14196
0.12621
0.11045
0.094686
0.078914
0.063137
0.047356
0.031572
0.015787
7.0888e-16

In order to reproduce this calculation with svds, do a QR decomposition of B. The singular values of the triangular matrix R are the same as for B.

[Q,R,p] = qr(B,0);

Plot the norm of each row of R.

rownormR = sqrt(diag(R*R'));
semilogy(rownormR)
hold on;
semilogy(size(R, 1), rownormR(end), 'ro')

The last entry in R is nearly zero, which causes instability in the solution.

Prevent this entry from corrupting the good parts of the solution by setting the last row of R to be exactly zero.

R(end,:) = 0;

Use svds to find the 10 smallest singular values of R. The results are comparable to those obtained by svd.

sr = svds(R,10,'smallest')
sr =

0.14196
0.12621
0.11045
0.094686
0.078914
0.063137
0.047356
0.031572
0.015787
0

To compute the singular vectors of B using this method, transform the left and right singular vectors using Q and the permutation vector p.

[U,S,V] = svds(R,20,'s');
U = Q*U;
V(p,:) = V;

## Input Arguments

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Input matrix. A is typically, but not always, a large and sparse matrix.

Data Types: double
Complex Number Support: Yes

Number of singular values to compute, specified as a positive scalar integer. svds returns fewer singular values than requested if either of these conditions are met:

• k is larger than min(m,n)

• sigma = 'smallestnz' and k is larger than the number of nonzero singular values of A

If k is too large, then svds replaces it with the maximum valid value of k.

Example: svds(A,2) returns the two largest singular values of A.

Type of singular values, specified as one of these values:

OptionDescription

'largest'

(default)
Largest singular values

'smallest'

Smallest singular values

'smallestnz

Smallest nonzero singular values

scalar

Singular values closest to a scalar shift

Example: svds(A,k,'smallest') computes the k smallest singular values.

Example: svds(A,k,100) computes the k singular values closest to 100.

Options structure, specified as a structure containing one or more of the fields in this table.

Option FieldDescriptionDefault
opts.tol

Convergence tolerance

1e-10
opts.maxit

Maximum number of iterations

100
opts.p

Maximum size of Krylov subspace

max(3*K,15)
opts.u0

Left initial starting vector

You can provide at most one of opts.u0 or opts.v0. If neither option is specified, then for an m-by-n matrix A, the default is:

• m < nopts.u0 = randn(m,1);

• m >= nopts.v0 = randn(n,1);

 Note:   svds selects u0 and v0 in a reproducible manner using a private random number stream. Changing the random number seed does not affect this use of randn.
opts.v0

Right initial starting vector

 Note:   svds ignores the option p when using a numeric scalar shift sigma.

Example: opts.tol = 1e-6, opts.maxit = 500 creates a structure with values set for the fields tol and maxit.

Data Types: struct

Matrix function, specified as a function handle. The function Afun must satisfy these conditions:

• Afun(x,'notransp') accepts a vector x and returns the product A*x.

• Afun(x,'transp') accepts a vector x and returns the product A'*x.

 Note:   Use function handles only in the case where sigma = 'largest' (which is the default).

Example: svds(Afun,[1000 1200])

Data Types: function_handle

Size of matrix A that is used by Afun, specified as a two-element size vector [m n].

Example: svds(Afun,[1000 1200])

## Output Arguments

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Singular values, returned as a column vector. The singular values are nonnegative real numbers listed in decreasing order.

Left singular vectors, returned as the columns of a matrix. If A is an m-by-n matrix and you request k singular values, then U is an m-by-k matrix with orthonormal columns.

Singular values, returned as a diagonal matrix. The diagonal elements of S are nonnegative singular values. If A is an m-by-n matrix and you request k singular values, then S is k-by-k.

Right singular vectors, returned as the columns of a matrix. If A is an m-by-n matrix and you request k singular values, then V is an n-by-k matrix with orthonormal columns.

Convergence flag, returned as a scalar. A value of 0 indicates that all the singular values converged. Otherwise, not all the singular values converged.

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### Tips

• svds generates the default starting vectors opts.u0 and opts.v0 using a private random number stream to ensure reproducibility across runs. Setting the random number generator state using rng before calling svds does not affect the output.

• Using svds is not the most efficient way to find a few singular values of small, dense matrices. If the problem fits into memory, it might be quicker to use svd(full(A)). For example, finding three singular values in a 500-by-500 matrix is a relatively small problem that is easily handled with svd.

• If svds fails to converge for a given matrix, increase the size of the Krylov subspace by increasing the value of opts.p. As secondary options, adjusting the maximum number of iterations, opts.maxit, and the convergence tolerance, opts.tol, can also help with convergence behavior.

• For faster performance, it sometimes works to increase k, especially when there are repeated singular values.

## References

[1] Baglama, J. and L. Reichel, "Augmented Implicitly Restarted Lanczos Bidiagonalization Methods." SIAM Journal on Scientific Computing. Vol. 27, 2005, pp. 19–42.

[2] Larsen, R. M. "Lanczos Bidiagonalization with partial reorthogonalization." Dept. of Computer Science, Aarhus University. DAIMI PB-357, 1998.