MATLAB: Is there a way to solve complex-valued non-linear equations

complexfsolvenonlinearnumeric equation solving

Hi guys,

I'm trying to solve a set of complex-valued non-linear equations, but I found that "fsolve" solves only real-valued equations. Is there another function that solves comlpex-valued equations? Or is there any way around it?

Thanks,

Best Answer

You can use fsolve. Split each complex-valued output into two outputs, the real() and imag().

You can do the same sort of thing for each complex-valued input.

Related Solutions

MATLAB: How to solve a complex-valued equation

So your equation is of the form

a / (g - x)^2 = k* x

Just multiply by (g - x)^2 in both sides, you'll get

a = (g - x)^2 * k* x
k*x^3 - 2*k*g*x^2 + k*g^2*x - a = 0

The 3 solutions (for each omega) can be obtained by

for k1 = 1:length(omega)
    omega_val = omega(k1);
    a = (1 / 2) * 1i * omega_val * (e_0 * A) * (V_dc + V_ac)^2;
    x = roots([k,-2*g*k,k*g^2,-a])
end

This gives you 3 solutions, I don't know which one you want to pick or not.

MATLAB: How to export complex-valued matrix to HDF file

Answer/summary after five years just for future reference

Charles' original question asked for a way to save complex matrices from MATLAB in HDF5 in such a way that Armadillo could read them. As a matter of fact, MAT-files v7.3 are indeed HDF5-based, and it happens that the way Armadillo stores complex matrices in HDF5 is the same as MATLAB's. So, you can just save one or more variables into a .mat file, making sure you're using the right version, and then load the variables in Armadillo just as if the .mat file was a regular HDF5 file (details and demonstration below).

As for the other way around, i.e. complex matrices from Armadillo to MATLAB, there is no native way to do it despite the format being basically the same. If you try to treat a HDF5 file generated from Armadillo as a .mat file, you get an error. However, per iakson's suggestions paid out, and EasyH5 (nightly as of 15 Dec 2019) now supports loading Armadillo's HDF5 file with complex matrices without extra effort.

Demo: from MATLAB to Armadillo and viceversa

You'll need:

MATLAB ≥ 2006b (or ≥ 2017a to save without compression) [source1, source2]
Armadillo ≥ 3.4 (bare minimum, highly discouraged, you cannot specify custom dataset names and you have to change the demo code); recommended ≥ 8.100 (demo code works without edits), highly suggested ≥ 8.300 (for .save() capabilities not covered in the demo, but useful in production) [source]. It's better to install Armadillo with the CMake installer so to generate the runtime wrapper library and make sure that HDF5 is enabled at Armadillo's build time; in this way, one can link with just '-larmadillo' and forget about HDF5 libraries forever.
EasyH5 Toolbox (https://github.com/fangq/easyh5), rev. ≥ 791dd7b (previous versions partially work as well; in previous versions you had to do some extra work to merge real and imaginary parts of Armadillo's complex arrays and the saveh5 function was not able to save in a format compatible with Armadillo, though you always had the .mat file option).

Let's first generate some complex data from MATLAB and let's save it in double-precision ASCII by separating real and imaginary parts, which can then be imported in Armadillo as a reference data, as HDF5 via EasyH5 and as compressed/uncompressed .mat file v7.3:

% Generate random data
myCx1000x1000Mat=randn(1000)+1i*randn(1000);
% Separate real/imag parts
myCx1000x1000Mat_Re=real(myCx1000x1000Mat);
myCx1000x1000Mat_Im=imag(myCx1000x1000Mat);
% Save as ASCII
save('myCx1000x1000Mat_Re.dat','myCx1000x1000Mat_Re','-ascii','-double');
save('myCx1000x1000Mat_Im.dat','myCx1000x1000Mat_Im','-ascii','-double');
% Save as MAT-file v7.3, both compressed and uncompressed
save('myCx1000x1000Mat_comp.mat','myCx1000x1000Mat','-v7.3');
save('myCx1000x1000Mat_nocomp.mat','myCx1000x1000Mat','-v7.3','-nocompression');
% Save as pure HDF5 via EasyH5, specifying a complex number storage format friendly to Armadillo
saveh5(myCx1000x1000Mat,'myCx1000x1000Mat.h5','ComplexFormat',{'real','imag'});

You can inspect the MAT-files and the HDF5 file via the h5disp function, e.g.:

h5disp('myCx1000x1000Mat_comp.mat')
h5disp('myCx1000x1000Mat_nocomp.mat')
h5disp('myCx1000x1000Mat.h5')

which outputs:

HDF5 myCx1000x1000Mat_comp.mat 
Group '/' 
    Dataset 'myCx1000x1000Mat' 
        Size:  1000x1000
        MaxSize:  InfxInf
        Datatype:   H5T_COMPOUND
            Member 'real':  H5T_IEEE_F64LE (double)
            Member 'imag':  H5T_IEEE_F64LE (double)
        ChunkSize:  4x1000
        Filters:  deflate(3)
        FillValue:  H5T_COMPOUND
        Attributes:
            'MATLAB_class':  'double'
HDF5 myCx1000x1000Mat_nocomp.mat 
Group '/' 
    Dataset 'myCx1000x1000Mat' 
        Size:  1000x1000
        MaxSize:  InfxInf
        Datatype:   H5T_COMPOUND
            Member 'real':  H5T_IEEE_F64LE (double)
            Member 'imag':  H5T_IEEE_F64LE (double)
        ChunkSize:  4x1000
        Filters:  deflate(0)
        FillValue:  H5T_COMPOUND
        Attributes:
            'MATLAB_class':  'double'
HDF5 myCx1000x1000Mat.h5 
Group '/' 
    Dataset 'myCx1000x1000Mat' 
        Size:  1000x1000
        MaxSize:  1000x1000
        Datatype:   H5T_COMPOUND
            Member 'real':  H5T_IEEE_F64LE (double)
            Member 'imag':  H5T_IEEE_F64LE (double)
        ChunkSize:  []
        Filters:  none
        FillValue:  H5T_COMPOUND

Now download the attached file testHDF5.txt, rename it as testHDF5.cpp and compile it with

g++ testHDF5.cpp -larmadillo -o testHDF5

This small C++ demo imports the data generated from MATLAB via the real/imag ASCII files and then compares the complex matrix imported in this way to:

The same matrix exported via Armadillo to a HDF5 file
The HDF5 file exported from MATLAB via the EasyH5 Toolbox
The non-compressed MAT-file
The compressed MAT-file

The output is the max difference between the ASCII-imported matrix and these matrices, in absolute value, which on my machine is:

matteo$ ./testHDF5

Max absolute difference between ASCII and load method:

armaHDF5 ---> (0,0)

HDF5 ---> (0,0)

MAT(NC) ---> (0,0)

MAT(C) ---> (0,0)

As you can see, the matrices are all equal to the reference matrix imported via ASCII files. The compressed MAT-file seems to work as well, though I would not blindly trust it; I suggest to either use the uncompressed version or the saveh5 function (unless you really have to save some space).

The C++ program also generates a 'myCx1000x1000Mat_ARMA.h5' file which contains the ASCII-imported matrix subsequently saved in HDF5 via Armadillo. We can check that the matrix saved in Armadillo is the same as the one that we have in MATLAB by first importing the matrix and then checking the max of the absolute difference:

myCx1000x1000Mat_ARMA=loadh5('myCx1000x1000Mat_ARMA.h5');
max(max(abs(myCx1000x1000Mat_ARMA.myCx1000x1000Mat-myCx1000x1000Mat)))

which returns exactly 0 on my machine, so the matrices are equal.

TL;DR

from MATLAB to Armadillo: save as a v7.3 MAT-file via the built in save() function with the appropriate options or as a pure HDF5 file via EasyH5's saveh5() function (in this case, you must use the syntax saveh5(...,'ComplexFormat',{'real','imag'}) ); then, in both cases, import the file in Armadillo as a regular HDF5 file.
from Armadillo to MATLAB: save as a HDF5 file from Armadillo and then import it into MATLAB by using EasyH5's loadh5() function, without the need to use additional options.

Best Answer

Related Solutions

MATLAB: How to solve a complex-valued equation

MATLAB: How to export complex-valued matrix to HDF file

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