MATLAB: How to invert a symbolic matrix bigger than 6-by-6

=a/bcomplexmatrixsymbolicSymbolic Math Toolboxx

How can I invert a symbolic matrix? How can I solve x = A\b where A is a symbolic matrix? Is it always possible?

For instance if I try to invert a matrix created as follows:

>> A = sym('a',6) MATLAB will find a solution.

However if A is defined as follows: >> A = sym('a',7)

MATLAB will not converge to a solution and keeps computing.

Best Answer

When it comes to solving symbolic matrices there are several methods that can be used such as:

Inverting matrix 'inv( )'
'linsolve( )'
'A\b'--> best solution for non-symbolic systems

There is a limitation in maths were you can not calculate matrices bigger than 6x6.

In fact, the symbolic linear solver can sometimes encounter a phenomenon known as "expression swell" where the size of numbers and expressions involved in the calculation grow dramatically as the calculation progresses.

You can find an example of this issue in the attached plot which shows the computational time needed to invert several symbolic square matrices of increasing order.

Please note that this issue is a symbolic maths limitation, which affects all the mathematical programs, it is not only a MATLAB issue.

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MATLAB: Backslash “\” operator is slow for symbolic matrices with diagonal numeric matrices

Notice from the example above that if we have a diagonal double matrix "A" and a symbolic matrix "B" then solving "A*x=B" takes longer if using "A\B" rather than "inv(A)*B". Though mathematically these two expressions are the same, the algorithms used to solve each is different. Because "A" is a double matrix and "B" is a symbolic matrix, using "inv(A)*B" first runs the "inv" function for doubles and then uses the "mtimes" function from the Symbolic Toolbox. However, when you use "A\B", the "mldivide" function from the symbolic toolbox is used which uses a linear solver to solve A*x = B.

As for exactly what causes the difference between these two algorithms, the symbolic linear solver can sometimes encounter a phenomenon known as "expression swell" where the size of numbers and expressions involved in the calculation grow dramatically as the calculation progresses. This is what causes the computation to take much longer. Expression swell can occur in exact computation, like that used in the Symbolic Toolbox, but not in floating-point computation like that used for double datatypes. It is likely that the diagonal nature of the double matrix leads to this issue.

To avoid this problem, use "inv(A)*B" rather than "A\B" when "A" is a diagonal double matrix and "B" is symbolic.

MATLAB: How to use custom date labels for the x-axis in MATLAB plots

It is more convenient to format the tick labels using the "XTick", "XTickLabelMode" and "XTickLabel" properties of the axis object (vs. using "datetick" function):

>> figure
>> box on
>>
>> % Start, end and number of ticks...
>> startDate = 7.3457e+05;
>> endDate = 7.3458e+05;
>> numbertick = ceil(endDate - startDate);
>>
>> % Plot them
>> xData = linspace(startDate, endDate, numbertick);
>> plot(xData,ones(size(xData)));
>>
>> %%Modify the axes properties
>> set(gca, 'XTick',xData)
>> set(gca, 'TickDir', 'out');
>> set(gca, 'XTickLabelMode', 'auto');
>>
>> % Modifying the labels
>> labels = get(gca, 'XTickLabel');
>>
>> for i = 1:length(labels)
>>     if i == 1 || i == ceil(length(labels)/2) || i == length(labels)
>>         labels{i} = datestr(xData(i), 'mm/dd');
>>     else
>>         labels{i} = '';
>>     end
>> end
>>
>> set(gca, 'XTickLabel', labels)

Best Answer

Related Solutions

MATLAB: Backslash “\” operator is slow for symbolic matrices with diagonal numeric matrices

MATLAB: How to use custom date labels for the x-axis in MATLAB plots

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