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
Functions that accept scalars or vectors and return matrices, such as
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
speye, respectively. There is no
sparse analog for the function
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
is a sparse vector. Columnwise functions such as
sum also return sparse vectors, even
though these vectors can be entirely nonzero. Important exceptions to this rule are
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
are full, while
S&F are sparse. In
some cases, the result might be sparse even though the matrix has few zero
Matrix concatenation using either the
cat function or square brackets produces sparse results for mixed
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
P*S or on the columns as
A permutation vector
p, which is a full vector containing a
1:n, acts on the rows of
S(p,:), or on the columns as
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
the permutation matrix
P. For example, the statements
P*S return the same matrix.
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
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
S*P' both produce
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
P is a sparse matrix, then both representations use storage
n and you can apply either to
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
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.
colperm computes a permutation that orders
the columns of a matrix by the number of nonzeros in each column from smallest to
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
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
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
You can change various parameters associated with details of the algorithms using the
Like the approximate minimum degree ordering, the nested dissection ordering algorithm
implemented by the
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 .
S is a sparse matrix, the following command returns three sparse
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
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
S. By itself,
U in a single matrix without the pivot
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
Q to improve sparsity in
You can control pivoting in sparse matrices using
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
thresh = 1 is the default. (The default
0.1 for the four-output
When you call
lu with three or less outputs, MATLAB automatically allocates the memory necessary to hold the sparse
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
colamd, then computing
lu(S(:,p)) takes less time and storage than computing
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 = 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 - 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,
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
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.
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')
S is a symmetric (or Hermitian), positive definite, sparse
matrix, the statement below returns a sparse, upper triangular matrix
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
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
chol, to calculate how much memory is allocated.
MATLAB computes the complete QR factorization of a sparse matrix
[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
Q'*B without computing
The Q-less QR factorization allows the solution of sparse least squares problems
with two steps:
[c,R] = qr(A,b); x = R\c
A is sparse, but not square, MATLAB uses these steps for the linear equation solving backslash operator:
x = A\b
It is also possible to solve a sequence of least squares linear systems with different
b, that are not necessarily known when
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
ichol functions provide
approximate, incomplete factorizations, which are useful as
preconditioners for sparse iterative methods.
ilu function produces three incomplete
lower-upper (ILU) factorizations: the zero-fill ILU
ILU(0)), a Crout version of ILU (
ILU with threshold dropping and pivoting (
ILU(0) never pivots and the resulting factors only have nonzeros in
positions where the input matrix had nonzeros. Both
ILUTP(tau), however, do threshold-based dropping with the
user-defined drop tolerance
A = gallery('neumann', 1600) + speye(1600);
ans = 7840
ans = 126478
A has 7840 nonzeros, and its complete LU factorization
has 126478 nonzeros. On the other hand, the following code shows the different ILU
[L,U] = ilu(A); nnz(L)+nnz(U)-size(A,1)
ans = 7840
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
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
ichol function provides zero-fill
incomplete Cholesky factorizations (
IC(0)) as well as
threshold-based dropping incomplete Cholesky factorizations (
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
A = delsq(numgrid('S',200)); nnz(A)
ans = 195228
ans = 7762589
Ahas 195228 nonzeros, and its complete Cholesky factorization without reordering has 7762589 nonzeros. By contrast:
L = ichol(A); nnz(L)
ans = 117216
ans = 3.5805e-17
opts.type = 'ict'; opts.droptol = 1e-4; L = ichol(A,opts); nnz(L)
ans = 1166754
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
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
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
specifies a preordering of the indices of
A, apply backslash to the
A(:,p), whose columns are the columns of
permuted according to the vector
x = A(:,p) \ b; x(p) = x; spparms(currentParms);
spparms(defaultParms) restores the controls to their
prior state, in case you use
A\b later without specifying an
Eleven functions are available that implement iterative methods for sparse systems of simultaneous linear systems.
Functions for Iterative Methods for Sparse Systems
Biconjugate gradient stabilized
|Biconjugate gradient stabilized (l)|
Conjugate gradient squared
Generalized minimum residual
Preconditioned conjugate gradient
|Transpose-free quasiminimal residual|
These methods are designed to solve Ax =
b or minimize the norm of b –
Ax. For the Preconditioned Conjugate Gradient
pcg, A must be a symmetric, positive
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
All eleven methods can make use of preconditioners. The linear system
is replaced by the equivalent system
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
L = ichol(A,struct('type','nofill','michol','on')); [x,flag,err,iter,res] = pcg(A,b,tol,maxit,L,L');
Two functions are available that compute a few specified eigenvalues or singular values.
svds is based on
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.
[V,lambda] = eigs(A,k,sigma)
k eigenvalues and corresponding eigenvectors of the matrix
A that are nearest the “shift”
sigma is omitted, the eigenvalues largest in magnitude are found. If
sigma is zero, the eigenvalues smallest in magnitude are found. A
B, can be included for the generalized eigenvalue problem:
Aυ = λBυ.
[U,S,V] = svds(A,k)
k largest singular values of
[U,S,V] = svds(A,k,'smallest')
k smallest singular values.
The numerical techniques used in
described in .
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.
ans = 1×2 2945 2945
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
 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.
 Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
 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.
 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.
 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.
 Saad, Yousef, Iterative Methods for Sparse Linear Equations. PWS Publishing Company, 1996.
 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.