The computational complexity of sparse operations is proportional to
`nnz`

, the number of nonzero elements in the matrix. Computational
complexity also depends linearly on the row size `m`

and column size
`n`

of the matrix, but is independent of the product
`m*n`

, the total number of zero and nonzero elements.

The complexity of fairly complicated operations, such as the solution of sparse linear equations, involves factors like ordering and fill-in, which are discussed in the previous section. In general, however, the computer time required for a sparse matrix operation is proportional to the number of arithmetic operations on nonzero quantities.

Sparse matrices propagate through computations according to these rules:

Functions that accept a matrix and return a scalar or constant-size vector always produce output in full storage format. For example, the

`size`

function always returns a full vector, whether its input is full or sparse.Functions that accept scalars or vectors and return matrices, such as

`zeros`

,`ones`

,`rand`

, and`eye`

, always return full results. This is necessary to avoid introducing sparsity unexpectedly. The sparse analog of`zeros(m,n)`

is simply`sparse(m,n)`

. The sparse analogs of`rand`

and`eye`

are`sprand`

and`speye`

, respectively. There is no sparse analog for the function`ones`

.Unary functions that accept a matrix and return a matrix or vector preserve the storage class of the operand. If

`S`

is a sparse matrix, then`chol(S)`

is also a sparse matrix, and`diag(S)`

is a sparse vector. Columnwise functions such as`max`

and`sum`

also return sparse vectors, even though these vectors can be entirely nonzero. Important exceptions to this rule are the`sparse`

and`full`

functions.Binary operators yield sparse results if both operands are sparse, and full results if both are full. For mixed operands, the result is full unless the operation preserves sparsity. If

`S`

is sparse and`F`

is full, then`S+F`

,`S*F`

, and`F\S`

are full, while`S.*F`

and`S&F`

are sparse. In some cases, the result might be sparse even though the matrix has few zero elements.Matrix concatenation using either the

`cat`

function or square brackets produces sparse results for mixed operands.

A permutation of the rows and columns of a sparse matrix `S`

can be
represented in two ways:

A permutation matrix

`P`

acts on the rows of`S`

as`P*S`

or on the columns as`S*P'`

.A permutation vector

`p`

, which is a full vector containing a permutation of`1:n`

, acts on the rows of`S`

as`S(p,:)`

, or on the columns as`S(:,p)`

.

For example:

p = [1 3 4 2 5] I = eye(5,5); P = I(p,:) e = ones(4,1); S = diag(11:11:55) + diag(e,1) + diag(e,-1)

p = 1 3 4 2 5 P = 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 S = 11 1 0 0 0 1 22 1 0 0 0 1 33 1 0 0 0 1 44 1 0 0 0 1 55

You can now try some permutations using the permutation vector `p`

and
the permutation matrix `P`

. For example, the statements
`S(p,:)`

and `P*S`

return the same matrix.

S(p,:)

ans = 11 1 0 0 0 0 1 33 1 0 0 0 1 44 1 1 22 1 0 0 0 0 0 1 55

P*S

ans = 11 1 0 0 0 0 1 33 1 0 0 0 1 44 1 1 22 1 0 0 0 0 0 1 55

Similarly, `S(:,p)`

and `S*P'`

both produce

S*P'

ans = 11 0 0 1 0 1 1 0 22 0 0 33 1 1 0 0 1 44 0 1 0 0 1 0 55

If `P`

is a sparse matrix, then both representations use storage
proportional to `n`

and you can apply either to `S`

in
time proportional to `nnz(S)`

. The vector representation is slightly more
compact and efficient, so the various sparse matrix permutation routines all return full row
vectors with the exception of the pivoting permutation in LU (triangular) factorization,
which returns a matrix compatible with the full LU factorization.

To convert between the two permutation representations:

n = 5; I = speye(n); Pr = I(p,:); Pc = I(:,p); pc = (1:n)*Pc

pc = 1 3 4 2 5

pr = (Pr*(1:n)')'

pr = 1 3 4 2 5

The inverse of `P`

is simply `R = P'`

. You can compute
the inverse of `p`

with `r(p) = 1:n`

.

r(p) = 1:5

r = 1 4 2 3 5

Reordering the columns of a matrix can often make its LU or QR factors sparser. Reordering the rows and columns can often make its Cholesky factors sparser. The simplest such reordering is to sort the columns by nonzero count. This is sometimes a good reordering for matrices with very irregular structures, especially if there is great variation in the nonzero counts of rows or columns.

The `colperm`

computes a permutation that orders
the columns of a matrix by the number of nonzeros in each column from smallest to
largest.

The reverse Cuthill-McKee ordering is intended to reduce the profile or bandwidth of
the matrix. It is not guaranteed to find the smallest possible bandwidth, but it usually
does. The `symrcm`

function actually operates on the
nonzero structure of the symmetric matrix `A + A'`

, but the result is
also useful for nonsymmetric matrices. This ordering is useful for matrices that come from
one-dimensional problems or problems that are in some sense *long and
thin*.

The degree of a node in a graph is the number of connections to that node. This is the same as the number of off-diagonal nonzero elements in the corresponding row of the adjacency matrix. The approximate minimum degree algorithm generates an ordering based on how these degrees are altered during Gaussian elimination or Cholesky factorization. It is a complicated and powerful algorithm that usually leads to sparser factors than most other orderings, including column count and reverse Cuthill-McKee. Because keeping track of the degree of each node is very time-consuming, the approximate minimum degree algorithm uses an approximation to the degree, rather than the exact degree.

These MATLAB^{®} functions implement the approximate minimum degree algorithm:

See Reordering and Factorization of Sparse Matrices for an example using
`symamd`

.

You can change various parameters associated with details of the algorithms using the
`spparms`

function.

For details on the algorithms used by `colamd`

and
`symamd`

, see [5]. The approximate degree the algorithms use is based
on [1].

Like the approximate minimum degree ordering, the nested dissection ordering algorithm
implemented by the `dissect`

function reorders the matrix rows and columns by considering the matrix to be the
adjacency matrix of a graph. The algorithm reduces the problem down to a much smaller
scale by collapsing together pairs of vertices in the graph. After reordering the small
graph, the algorithm then applies projection and refinement steps to expand the graph back
to the original size.

The nested dissection algorithm produces high quality reorderings and performs particularly well with finite element matrices compared to other reordering techniques. For more information about the nested dissection ordering algorithm, see [7].

If `S`

is a sparse matrix, the following command returns three sparse
matrices `L`

, `U`

, and `P`

such that
`P*S = L*U`

.

[L,U,P] = lu(S);

`lu`

obtains the factors by Gaussian elimination with partial
pivoting. The permutation matrix `P`

has only `n`

nonzero elements. As with dense matrices, the statement `[L,U] = lu(S)`

returns a permuted unit lower triangular matrix and an upper triangular matrix whose
product is `S`

. By itself, `lu(S)`

returns
`L`

and `U`

in a single matrix without the pivot
information.

The three-output syntax `[L,U,P] = lu(S)`

selects
`P`

via numerical partial pivoting, but does not pivot to improve
sparsity in the `LU`

factors. On the other hand, the four-output syntax
`[L,U,P,Q] = lu(S)`

selects `P`

via threshold partial
pivoting, and selects `P`

and `Q`

to improve sparsity in
the `LU`

factors.

You can control pivoting in sparse matrices using

lu(S,thresh)

where `thresh`

is a pivot threshold in [0,1]. Pivoting occurs when
the diagonal entry in a column has magnitude less than `thresh`

times the
magnitude of any sub-diagonal entry in that column. `thresh = 0`

forces
diagonal pivoting. `thresh = 1`

is the default. (The default
for `thresh`

is `0.1`

for the four-output
syntax).

When you call `lu`

with three or less outputs, MATLAB automatically allocates the memory necessary to hold the sparse
`L`

and `U`

factors during the factorization. Except
for the four-output syntax, MATLAB does not use any symbolic LU prefactorization to determine the memory
requirements and set up the data structures in advance.

**Reordering and Factorization of Sparse Matrices**

This example shows the effects of reordering and factorization on sparse matrices.

If you obtain a good column permutation `p`

that reduces fill-in, perhaps from `symrcm`

or `colamd`

, then computing `lu(S(:,p))`

takes less time and storage than computing `lu(S)`

.

Create a sparse matrix using the Bucky ball example.

B = bucky;

`B`

has exactly three nonzero elements in each row and column.

Create two permutations, `r`

and `m`

using `symrcm`

and `symamd`

respectively.

r = symrcm(B); m = symamd(B);

The two permutations are the symmetric reverse Cuthill-McKee ordering and the symmetric approximate minimum degree ordering.

Create spy plots to show the three adjacency matrices of the Bucky Ball graph with these three different numberings. The local, pentagon-based structure of the original numbering is not evident in the others.

figure subplot(1,3,1) spy(B) title('Original') subplot(1,3,2) spy(B(r,r)) title('Reverse Cuthill-McKee') subplot(1,3,3) spy(B(m,m)) title('Min Degree')

The reverse Cuthill-McKee ordering, `r`

, reduces the bandwidth and concentrates all the nonzero elements near the diagonal. The approximate minimum degree ordering, `m`

, produces a fractal-like structure with large blocks of zeros.

To see the fill-in generated in the LU factorization of the Bucky ball, use `speye`

, the sparse identity matrix, to insert -3s on the diagonal of `B`

.

B = B - 3*speye(size(B));

Since each row sum is now zero, this new `B`

is actually singular, but it is still instructive to compute its LU factorization. When called with only one output argument, `lu`

returns the two triangular factors, `L`

and `U`

, in a single sparse matrix. The number of nonzeros in that matrix is a measure of the time and storage required to solve linear systems involving `B`

.

Here are the nonzero counts for the three permutations being considered.

`lu(B)`

(Original): 1022`lu(B(r,r))`

(Reverse Cuthill-McKee): 968`lu(B(m,m))`

(Approximate minimum degree): 636

Even though this is a small example, the results are typical. The original numbering scheme leads to the most fill-in. The fill-in for the reverse Cuthill-McKee ordering is concentrated within the band, but it is almost as extensive as the first two orderings. For the approximate minimum degree ordering, the relatively large blocks of zeros are preserved during the elimination and the amount of fill-in is significantly less than that generated by the other orderings.

The `spy`

plots below reflect the characteristics of each reordering.

figure subplot(1,3,1) spy(lu(B)) title('Original') subplot(1,3,2) spy(lu(B(r,r))) title('Reverse Cuthill-McKee') subplot(1,3,3) spy(lu(B(m,m))) title('Min Degree')

If `S`

is a symmetric (or Hermitian), positive definite, sparse
matrix, the statement below returns a sparse, upper triangular matrix `R`

so that `R'*R = S`

.

R = chol(S)

`chol`

does not automatically pivot for sparsity, but you can compute
approximate minimum degree and profile limiting permutations for use with
`chol(S(p,p))`

.

Since the Cholesky algorithm does not use pivoting for sparsity and does not require
pivoting for numerical stability, `chol`

does a quick calculation of the
amount of memory required and allocates all the memory at the start of the factorization.
You can use `symbfact`

, which uses the same algorithm
as `chol`

, to calculate how much memory is allocated.

MATLAB computes the complete QR factorization of a sparse matrix
`S`

with

[Q,R] = qr(S)

[Q,R,E] = qr(S)

but this is often impractical. The unitary matrix `Q`

often fails to
have a high proportion of zero elements. A more practical alternative, sometimes known as
“the Q-less QR factorization,” is available.

With one sparse input argument and one output argument

R = qr(S)

returns just the upper triangular portion of the QR factorization. The matrix
`R`

provides a Cholesky factorization for the matrix associated with
the normal equations:

R'*R = S'*S

However, the loss of numerical information inherent in the computation of
`S'*S`

is avoided.

With two input arguments having the same number of rows, and two output arguments, the statement

[C,R] = qr(S,B)

applies the orthogonal transformations to `B`

, producing ```
C =
Q'*B
```

without computing `Q`

.

The Q-less QR factorization allows the solution of sparse least squares problems

$$\text{minimize}{\Vert Ax-b\Vert}_{2}$$

with two steps:

[c,R] = qr(A,b); x = R\c

If `A`

is sparse, but not square, MATLAB uses these steps for the linear equation solving backslash operator:

x = A\b

Or, you can do the factorization yourself and examine `R`

for rank deficiency.

It is also possible to solve a sequence of least squares linear systems with different
right-hand sides, `b`

, that are not necessarily known when ```
R =
qr(A)
```

is computed. The approach solves the “semi-normal equations
`R'*R*x = A'*b`

with

x = R\(R'\(A'*b))

and then employs one step of iterative refinement to reduce round off error:

r = b - A*x; e = R\(R'\(A'*r)); x = x + e

The `ilu`

and `ichol`

functions provide
approximate,* incomplete *factorizations, which are useful as
preconditioners for sparse iterative methods.

The `ilu`

function produces three *incomplete
lower-upper* (ILU) factorizations: the *zero-fill ILU*
(`ILU(0)`

), a Crout version of ILU (`ILUC(tau)`

), and
ILU with threshold dropping and pivoting (`ILUTP(tau)`

). The
`ILU(0)`

never pivots and the resulting factors only have nonzeros in
positions where the input matrix had nonzeros. Both `ILUC(tau)`

and
`ILUTP(tau)`

, however, do threshold-based dropping with the
user-defined drop tolerance `tau`

.

For example:

`A = gallery('neumann', 1600) + speye(1600);`

ans = 7840

nnz(lu(A))

ans = 126478

shows that `A`

has 7840 nonzeros, and its complete LU factorization
has 126478 nonzeros. On the other hand, the following code shows the different ILU
outputs:

[L,U] = ilu(A); nnz(L)+nnz(U)-size(A,1)

ans = 7840

norm(A-(L*U).*spones(A),'fro')./norm(A,'fro')

ans = 4.8874e-17

```
opts.type = 'ilutp';
opts.droptol = 1e-4;
[L,U,P] = ilu(A, opts);
nnz(L)+nnz(U)-size(A,1)
```

ans = 31147

norm(P*A - L*U,'fro')./norm(A,'fro')

ans = 9.9224e-05

```
opts.type = 'crout';
[L,U,P] = ilu(A, opts);
nnz(L)+nnz(U)-size(A,1)
```

ans = 31083

norm(P*A-L*U,'fro')./norm(A,'fro')

ans = 9.7344e-05

These calculations show that the zero-fill factors have 7840 nonzeros, the
`ILUTP(1e-4)`

factors have 31147 nonzeros, and the
`ILUC(1e-4)`

factors have 31083 nonzeros. Also, the relative error of
the product of the zero-fill factors is essentially zero on the pattern of
`A`

. Finally, the relative error in the factorizations produced with
threshold dropping is on the same order of the drop tolerance, although this is not
guaranteed to occur. See the `ilu`

reference page for more options and
details.

The `ichol`

function provides *zero-fill
incomplete Cholesky factorizations* (`IC(0)`

) as well as
threshold-based dropping incomplete Cholesky factorizations (`ICT(tau)`

)
of symmetric, positive definite sparse matrices. These factorizations are the analogs of
the incomplete LU factorizations above and have many of the same characteristics. For
example:

```
A = delsq(numgrid('S',200));
nnz(A)
```

ans = 195228

`nnz(chol(A,'lower'))`

ans = 7762589

`A`

has 195228 nonzeros, and its complete Cholesky factorization
without reordering has 7762589 nonzeros. By
contrast:L = ichol(A); nnz(L)

ans = 117216

norm(A-(L*L').*spones(A),'fro')./norm(A,'fro')

ans = 3.5805e-17

```
opts.type = 'ict';
opts.droptol = 1e-4;
L = ichol(A,opts);
nnz(L)
```

ans = 1166754

norm(A-L*L','fro')./norm(A,'fro')

ans = 2.3997e-04

`IC(0)`

has nonzeros only in the pattern of the lower triangle of
`A`

, and on the pattern of `A`

, the product of the
factors matches. Also, the `ICT(1e-4)`

factors are considerably sparser
than the complete Cholesky factor, and the relative error between `A`

and` L*L'`

is on the same order of the drop tolerance. It is important
to note that unlike the factors provided by `chol`

, the default factors
provided by `ichol`

are lower triangular. See the `ichol`

reference page for more information.

There are two different classes of methods for solving systems of simultaneous linear equations:

*Direct methods*are usually variants of Gaussian elimination. These methods involve the individual matrix elements directly, through matrix operations such as LU or Cholesky factorization. MATLAB implements direct methods through the matrix division operators / and \, which you can use to solve linear systems.*Iterative methods*produce only an approximate solution after a finite number of steps. These methods involve the coefficient matrix only indirectly, through a matrix-vector product or an abstract linear operator. Iterative methods are usually applied only to sparse matrices.

Direct methods are usually faster and more generally applicable than indirect methods, if there is enough storage available to carry them out. Iterative methods are usually applicable to restricted cases of equations and depend on properties like diagonal dominance or the existence of an underlying differential operator. Direct methods are implemented in the core of the MATLAB software and are made as efficient as possible for general classes of matrices. Iterative methods are usually implemented in MATLAB-language files and can use the direct solution of subproblems or preconditioners.

**Using a Different Preordering. **If `A`

is not diagonal, banded, triangular, or a permutation of a
triangular matrix, backslash (\) reorders the indices of `A`

to reduce
the amount of fill-in—that is, the number of nonzero entries that are added to
the sparse factorization matrices. The new ordering, called a
*preordering*, is performed before the factorization of
`A`

. In some cases, you might be able to provide a better preordering
than the one used by the backslash algorithm.

To use a different preordering, first turn off both of the automatic preorderings
that backslash might perform by default, using the function `spparms`

as follows:

defaultParms = spparms; spparms('autoamd',0); spparms('autommd',0);

Now, assuming you have created a permutation vector `p`

that
specifies a preordering of the indices of `A`

, apply backslash to the
matrix `A(:,p)`

, whose columns are the columns of `A`

,
permuted according to the vector `p`

.

x = A(:,p) \ b; x(p) = x; spparms(currentParms);

The command `spparms(defaultParms)`

restores the controls to their
prior state, in case you use `A\b`

later without specifying an
appropriate preordering.

Eleven functions are available that implement iterative methods for sparse systems of simultaneous linear systems.

**Functions for Iterative Methods for Sparse Systems**

Function |
Method |
---|---|

Biconjugate gradient | |

Biconjugate gradient stabilized | |

`bicgstabl` | Biconjugate gradient stabilized (l) |

Conjugate gradient squared | |

Generalized minimum residual | |

Least squares | |

Minimum residual | |

Preconditioned conjugate gradient | |

Quasiminimal residual | |

Symmetric LQ | |

`tfqmr` | Transpose-free quasiminimal residual |

These methods are designed to solve *A**x* =
*b* or minimize the norm of *b* –
*A**x*. For the Preconditioned Conjugate Gradient
method, `pcg`

, *A* must be a symmetric, positive
definite matrix. `minres`

and `symmlq`

can be used on
symmetric indefinite matrices. For `lsqr`

, the matrix need not be square.
The other seven can handle nonsymmetric, square matrices and each method has a distinct
benefit.

All eleven methods can make use of preconditioners. The linear system

$$Ax=b$$

is replaced by the equivalent system

$${M}^{-1}Ax={M}^{-1}b$$

The preconditioner *M* is chosen to accelerate convergence of the
iterative method. In many cases, the preconditioners occur naturally in the mathematical
model. A partial differential equation with variable coefficients can be approximated by
one with constant coefficients, for example. Incomplete matrix factorizations can be used
in the absence of natural preconditioners.

The five-point finite difference approximation to Laplace's equation on a square,
two-dimensional domain provides an example. The following statements use the
preconditioned conjugate gradient method preconditioner
*M* = *L***L*', where
*L* is the zero-fill incomplete Cholesky factor of
*A*.

```
A = delsq(numgrid('S',50));
b = ones(size(A,1),1);
tol = 1e-3;
maxit = 100;
L = ichol(A);
[x,flag,err,iter,res] = pcg(A,b,tol,maxit,L,L');
```

Twenty-one iterations are required to achieve the prescribed accuracy. On the other
hand, using a different preconditioner may yield better results. For example, using
`ichol`

to construct a modified incomplete Cholesky, the prescribed
accuracy is met after only 15
iterations:

L = ichol(A,struct('type','nofill','michol','on')); [x,flag,err,iter,res] = pcg(A,b,tol,maxit,L,L');

Background information on these iterative methods and incomplete factorizations is available in [2] and [6].

Two functions are available that compute a few specified eigenvalues or singular values.
`svds`

is based on `eigs`

.

**Functions to Compute a Few Eigenvalues or Singular Values**

These functions are most frequently used with sparse matrices, but they can be used with full matrices or even with linear operators defined in MATLAB code.

The statement

[V,lambda] = eigs(A,k,sigma)

finds the `k`

eigenvalues and corresponding eigenvectors of the matrix
`A`

that are nearest the “shift” `sigma`

.
If `sigma`

is omitted, the eigenvalues largest in magnitude are found. If
`sigma`

is zero, the eigenvalues smallest in magnitude are found. A
second matrix, `B`

, can be included for the generalized eigenvalue problem:
Aυ = λBυ.

The statement

[U,S,V] = svds(A,k)

finds the `k`

largest singular values of `A`

and

```
[U,S,V] = svds(A,k,'smallest')
```

finds the `k`

smallest singular values.

The numerical techniques used in `eigs`

and `svds`

are
described in [6].

This example shows how to find the smallest eigenvalue and eigenvector of a sparse matrix.

Set up the five-point Laplacian difference operator on a 65-by-65 grid in an *L*-shaped, two-dimensional domain.

```
L = numgrid('L',65);
A = delsq(L);
```

Determine the order and number of nonzero elements.

size(A)

`ans = `*1×2*
2945 2945

nnz(A)

ans = 14473

`A`

is a matrix of order 2945 with 14,473 nonzero elements.

Compute the smallest eigenvalue and eigenvector.

`[v,d] = eigs(A,1,'smallestabs');`

Distribute the components of the eigenvector over the appropriate grid points and produce a contour plot of the result.

```
L(L>0) = full(v(L(L>0)));
x = -1:1/32:1;
contour(x,x,L,15)
axis square
```

[1] Amestoy, P. R., T. A. Davis, and I. S. Duff, “An
Approximate Minimum Degree Ordering Algorithm,” *SIAM Journal on Matrix
Analysis and Applications*, Vol. 17, No. 4, Oct. 1996, pp. 886-905.

[2] Barrett, R., M. Berry, T. F. Chan, et al.,
*Templates for the Solution of Linear Systems: Building Blocks for Iterative
Methods*, SIAM, Philadelphia, 1994.

[3] Davis, T.A., Gilbert, J. R., Larimore, S.I., Ng, E., Peyton,
B., “A Column Approximate Minimum Degree Ordering Algorithm,”
*Proc. SIAM Conference on Applied Linear Algebra*, Oct. 1997,
p. 29.

[4] Gilbert, John R., Cleve Moler, and Robert Schreiber,
“Sparse Matrices in MATLAB: Design and Implementation,” *SIAM J.
Matrix Anal. Appl*., Vol. 13, No. 1, January 1992, pp. 333-356.

[5] Larimore, S. I., *An Approximate Minimum Degree
Column Ordering Algorithm*, MS Thesis, Dept. of Computer and Information
Science and Engineering, University of Florida, Gainesville, FL, 1998.

[6] Saad, Yousef, *Iterative Methods for Sparse Linear
Equations*. PWS Publishing Company, 1996.

[7] Karypis, George and Vipin
Kumar. "A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs."
*SIAM Journal on Scientific Computing*. Vol. 20, Number 1, 1999,
pp. 359–392.