3-Tuple storage

>> sparse(diag(0:3))
ans =

   (2,2)        1
   (3,3)        2
   (4,4)        3

As is evident from the display format for a sparse matrix, the non-zero values in the matrix are stored as (i,j,value) tuples. The zero elements are implied — anything not stored is treated as a zero.

The find function and the sparse function work directly with these tuples

• [i,j,v] = find(S) returns the truple row, column and value values as separate arrays.
• S = sparse(i,j,v) reconstructs the sparse array from the tuple values.

and can be used to switch back and forth between the tuple and the sparse matrix. The tuple format is a particularly compact way of representing sparse data and can be easily written to files that can be read by MATLAB and other data analysis packages.

Column oriented storage

The sparse storage format is optimized for column-oriented operations which can be important to understand when crafting m-files that use the sparse matrix. This can be especially true when working with huge matrices where the orientation of a matrix has a big effect on processing speed or memory requirements. The following figure illustrates the sparse storage approach used within MATLAB.

A column spine contains pointers to the elements in each column. In addition, the elements in the column are stored in row sorted order to improve the search time for a particular (row,col) element. Columns without elements contain a pointer to an empty column list.

This has several implications when working with the sparse matrix.

1. The column size of the matrix size(S,2) is significant since a column pointer is created for a column even if there are no elements in that column.
2. The row size of the matrix size(S,1) has no impact on the amount of memory used by the sparse matrix.
3. Rows that contain only zeros take up no space.
4. Accessing the columns of a sparse matrix is faster than accessing the rows of a sparse matrix.

The ramifications of these implications are that we should prefer algorithms that operate on the columns of sparse matrices and that orient the matrix so that there are less columns than rows.

¹You can use the whos command to determine the actual memory allocated to a sparse matrix.

When used with the tuple syntax, sparse(i,j,v) or sparse(i,j,v,m,n), any repeated (i,j) elements are automatically accumulated! In other words, the sparse function automatically sums any values that are assigned to the same location.

This is in contrast to normal array assignment where a statement like

A(i+(j-1)*m) = v;

(for a full array A) overwrites any repeated elements (leaving only the last value assigned in that spot).

The accumulation behavior can be used in some cases to enhance performance. For example, the following code uses sparse to compute the image histogram for an indexed image (the code also displays the histogram counts as a bar chart).

% Load one of the MATLAB demo images.
% The mat-file contains:
%   X   - the index image matrix
%   map - its colormap
load durer
counts = sparse(1,X,1);
bar(counts)

By mapping all the image values to a single row, the accumulation feature of sparse automatically computes the histogram counts!

accumarray

Current versions of MATLAB also contain a function called accumarray that performs the accumulation trick for non-sparse matrices (and you can change the accumulation function too). The accumarray function is even more powerful than sparse because it includes options to set the fill value to something other than zero and it handles cell arrays too.

Typical applications of mathematical engineering involve mathematical analysis, mathematical modeling, simulations and visualization techniques (including creating specialized user interfaces or GUI’s) that help engineers and researchers do their jobs.

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Dr. Clay M. Thomposn, founder Mathematical Engineering

Dr. Clay M. Thompson is one of the world’s foremost MATLAB programmers. He knows MATLAB inside and out, and specializes in using MATLAB to develop cost-effective, custom, proprietary applications for our clients.

Dr. Thompson worked at The MathWorks, Inc. (the makers of MATLAB) for 9 years developing toolboxes and writing m-files. He was the chief language designer for MATLAB 5.0 (the version of MATLAB that introduced multi-dimensional arrays, cell arrays, and MATLAB objects). In 1999, he left The MathWorks and began providing MATLAB consulting services.

Over the years, Dr. Thompson has written thousands of m- and MEX-files. If you use MATLAB, you have probably used many of the m- and mex-files he created. He is the author of the Image Processing Toolbox and a co-author of the Control Toolbox as well as many of the functions in the MATLAB toolbox itself. His intimate knowledge of MATLAB allows him to produce optimized m-files that run very fast.

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Our principal, Dr. Clay Thompson, has been optimizing m-files for years, often achieving a speed improvement for a collection of m-files of 400% or more. In some cases, we have even increased performance by 4000%!

There are a few requirements. If your problem fits those requirements, just contact us and let us know you want to take advantage of the performance optimization service. If your problem is too big, uses other toolboxes or doesn’t meet the requirements below, then contact us for a no-obligation estimate.

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