c++,matlab,linear-algebra,eigen,intel-mkl , Rewriting Matlab eig(A,B) (Generalized eigenvalues/eigenvectors) to C/C++


Rewriting Matlab eig(A,B) (Generalized eigenvalues/eigenvectors) to C/C++

Question:

Tag: c++,matlab,linear-algebra,eigen,intel-mkl

Do anyone have any idea how can I rewrite eig(A,B) from Matlab used to calculate generalized eigenvector/eigenvalues? I've been struggling with this problem lately. So far:

Matlab definition of eig function I need:

[V,D] = eig(A,B) produces a diagonal matrix D of generalized
eigenvalues and a full matrix V whose columns are the corresponding
eigenvectors so that A*V = B*V*D.
  1. So far I tried the Eigen library (http://eigen.tuxfamily.org/dox/classEigen_1_1GeneralizedSelfAdjointEigenSolver.html)

My implementation looks like this:

std::pair<Matrix4cd, Vector4d> eig(const Matrix4cd& A, const Matrix4cd& B)
{
    Eigen::GeneralizedSelfAdjointEigenSolver<Matrix4cd> solver(A, B);

    Matrix4cd V = solver.eigenvectors();
    Vector4d D = solver.eigenvalues();

    return std::make_pair(V, D);
}

But first thing that comes to my mind is, that I can't use Vector4cd as .eigenvalues() doesn't return complex values where Matlab does. Furthermore results of .eigenvectors() and .eigenvalues() for the same matrices are not the same at all:

C++:

Matrix4cd x;
Matrix4cd y;
pair<Matrix4cd, Vector4d> result;

for (int i = 0; i < 4; i++)
{
    for (int j = 0; j < 4; j++)
    {
        x.real()(i,j) = (double)(i+j+1+i*3);
        y.real()(i,j) = (double)(17 - (i+j+1+i*3));

        x.imag()(i,j) = (double)(i+j+1+i*3);
        y.imag()(i,j) = (double)(17 - (i+j+1+i*3));
    }
}
result = eig(x,y);
cout << result.first << endl << endl;
cout << result.second << endl << endl;

Matlab:

for i=1:1:4
    for j=1:1:4
        x(i,j) = complex((i-1)+(j-1)+1+((i-1)*3), (i-1)+(j-1)+1+((i-1)*3));
        y(i,j) = complex(17 - ((i-1)+(j-1)+1+((i-1)*3)), 17 - ((i-1)+(j-1)+1+((i-1)*3)));
    end
end

[A,B] = eig(x,y)

So I give eig the same 4x4 matrices holding values 1-16 ascending (x) and descending (y). But I receive different results, furthermore Eigen method returns double from eigenvalues while Matlab returns complex dobule. I also find out that there is other Eigen solver named GeneralizedEigenSolver. That one in the documentation (http://eigen.tuxfamily.org/dox/classEigen_1_1GeneralizedEigenSolver.html) has written that it solves A*V = B*V*D but to be honest I tried it and results (matrix sizes) are not the same size as Matlab so I got quite lost how it works (examplary results are on the website I've linked). It also has only .eigenvector method.

C++ results:

(-0.222268,-0.0108754) (0.0803437,-0.0254809) (0.0383264,-0.0233819) (0.0995482,0.00682079)
(-0.009275,-0.0182668) (-0.0395551,-0.0582127) (0.0550395,0.03434) (-0.034419,-0.0287563)
(-0.112716,-0.0621061) (-0.010788,0.10297) (-0.0820552,0.0294896) (-0.114596,-0.146384)
(0.28873,0.257988) (0.0166259,-0.0529934) (0.0351645,-0.0322988) (0.405394,0.424698)

-1.66983
-0.0733194
0.0386832
3.97933

Matlab results:

[A,B] = eig(x,y)


A =

  Columns 1 through 3

  -0.9100 + 0.0900i  -0.5506 + 0.4494i   0.3614 + 0.3531i
   0.7123 + 0.0734i   0.4928 - 0.2586i  -0.5663 - 0.4337i
   0.0899 - 0.4170i  -0.1210 - 0.3087i   0.0484 - 0.1918i
   0.1077 + 0.2535i   0.1787 + 0.1179i   0.1565 + 0.2724i

  Column 4

  -0.3237 - 0.3868i
   0.2338 + 0.7662i
   0.5036 - 0.3720i
  -0.4136 - 0.0074i


B =

  Columns 1 through 3

  -1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -1.0000 - 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -4.5745 - 1.8929i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i

  Column 4

   0.0000 + 0.0000i
   0.0000 + 0.0000i
   0.0000 + 0.0000i
  -0.3317 + 1.1948i
  1. Second try was with Intel IPP but it seems that it solves only A*V = V*D and support told me that it's not supported anymore.

https://software.intel.com/en-us/node/505270 (list of constructors for Intel IPP)

  1. I got suggestion to move from Intel IPP to MKL. I did it and hit the wall again. I tried to check all algorithms for Eigen but it seems that there are only A*V = V*D problems solved. I was checking lapack95.lib. The list of algorithms used by this library is available there: https://software.intel.com/sites/products/documentation/doclib/mkl_sa/11/mkl_lapack_examples/index.htm#dsyev.htm

Somewhere on the web I could find topic on Mathworks when someone said that managed to solve my problem partially with usage of MKL:

http://jp.mathworks.com/matlabcentral/answers/40050-generalized-eigenvalue-and-eigenvectors-differences-between-matlab-eig-a-b-and-mkl-lapack-dsygv

Person said that he/she used dsygv algorithm but I can't locate anything like that on the web. Maybe it's a typo.

Anyone has any other proposition/idea how can I implement it? Or maybe point my mistake. I'd appreciate that.


EDIT: In comments I've received a hint that I was using Eigen solver wrong. My A matrix wasn't self-adjoint and my B matrix wasn't positive-definite. I took matrices from program I want to rewrite to C++ (from random iteration) and checked if they meet the requirements. They did:

Rj =

  1.0e+02 *

 Columns 1 through 3

   0.1302 + 0.0000i  -0.0153 + 0.0724i   0.0011 - 0.0042i
  -0.0153 - 0.0724i   1.2041 + 0.0000i  -0.0524 + 0.0377i
   0.0011 + 0.0042i  -0.0524 - 0.0377i   0.0477 + 0.0000i
  -0.0080 - 0.0108i   0.0929 - 0.0115i  -0.0055 + 0.0021i

  Column 4

  -0.0080 + 0.0108i
   0.0929 + 0.0115i
  -0.0055 - 0.0021i
   0.0317 + 0.0000i

Rt =

  Columns 1 through 3

   4.8156 + 0.0000i  -0.3397 + 1.3502i  -0.2143 - 0.3593i
  -0.3397 - 1.3502i   7.3635 + 0.0000i  -0.5539 - 0.5176i
  -0.2143 + 0.3593i  -0.5539 + 0.5176i   1.7801 + 0.0000i
   0.5241 + 0.9105i   0.9514 + 0.6572i  -0.7302 + 0.3161i

  Column 4

   0.5241 - 0.9105i
   0.9514 - 0.6572i
  -0.7302 - 0.3161i
   9.6022 + 0.0000i

As for Rj which is now my A - it is self-adjoint because Rj = Rj' and Rj = ctranspose(Rj). (http://mathworld.wolfram.com/Self-AdjointMatrix.html)

As for Rt which is now my B - it is Positive-Definite what is checked with method linked to me. (http://www.mathworks.com/matlabcentral/answers/101132-how-do-i-determine-if-a-matrix-is-positive-definite-using-matlab). So

>> [~,p] = chol(Rt)

p =

     0

I've rewritten matrices manually to C++ and performed eig(A,B) again with matrices meeting requirements:

Matrix4cd x;
Matrix4cd y;
pair<Matrix4cd, Vector4d> result;

x.real()(0,0) = 13.0163601949795;
x.real()(0,1) = -1.53172561296005;
x.real()(0,2) = 0.109594869350436;
x.real()(0,3) = -0.804231869422614;

x.real()(1,0) = -1.53172561296005;
x.real()(1,1) = 120.406645675346;
x.real()(1,2) = -5.23758765476463;
x.real()(1,3) = 9.28686785230169;

x.real()(2,0) = 0.109594869350436; 
x.real()(2,1) = -5.23758765476463;
x.real()(2,2) = 4.76648319080400;
x.real()(2,3) = -0.552823839520508;

x.real()(3,0) = -0.804231869422614;
x.real()(3,1) = 9.28686785230169;
x.real()(3,2) = -0.552823839520508;
x.real()(3,3) = 3.16510496622613;

x.imag()(0,0) = -0.00000000000000;
x.imag()(0,1) = 7.23946944213164;
x.imag()(0,2) = 0.419181335323979;
x.imag()(0,3) = 1.08441894337449;

x.imag()(1,0) = -7.23946944213164;
x.imag()(1,1) = -0.00000000000000;
x.imag()(1,2) = 3.76849276970080;
x.imag()(1,3) = 1.14635625342266;

x.imag()(2,0) = 0.419181335323979;
x.imag()(2,1) = -3.76849276970080;
x.imag()(2,2) = -0.00000000000000;
x.imag()(2,3) = 0.205129702522089;

x.imag()(3,0) = -1.08441894337449;
x.imag()(3,1) = -1.14635625342266;
x.imag()(3,2) = 0.205129702522089;
x.imag()(3,3) = -0.00000000000000;

y.real()(0,0) = 4.81562784930907;
y.real()(0,1) = -0.339731222392148;
y.real()(0,2) = -0.214319720979258;
y.real()(0,3) = 0.524107127885349;

y.real()(1,0) = -0.339731222392148;
y.real()(1,1) = 7.36354235698375;
y.real()(1,2) = -0.553927983436786;
y.real()(1,3) = 0.951404408649307;

y.real()(2,0) = -0.214319720979258;
y.real()(2,1) = -0.553927983436786;
y.real()(2,2) = 1.78008768533745;
y.real()(2,3) = -0.730246631850385;

y.real()(3,0) = 0.524107127885349;
y.real()(3,1) = 0.951404408649307;
y.real()(3,2) = -0.730246631850385;
y.real()(3,3) = 9.60215057284395;

y.imag()(0,0) = -0.00000000000000;
y.imag()(0,1) = 1.35016928394966;
y.imag()(0,2) = -0.359262708214312;
y.imag()(0,3) = -0.910512495060186;

y.imag()(1,0) = -1.35016928394966;
y.imag()(1,1) = -0.00000000000000;
y.imag()(1,2) = -0.517616473138836;
y.imag()(1,3) = -0.657235460367660;

y.imag()(2,0) = 0.359262708214312;
y.imag()(2,1) = 0.517616473138836;
y.imag()(2,2) = -0.00000000000000;
y.imag()(2,3) = -0.316090662865005;

y.imag()(3,0) = 0.910512495060186;
y.imag()(3,1) = 0.657235460367660;
y.imag()(3,2) = 0.316090662865005;
y.imag()(3,3) = -0.00000000000000;

result = eig(x,y);

cout << result.first << endl << endl;
cout << result.second << endl << endl;

And the results of C++:

(0.0295948,0.00562174) (-0.253532,0.0138373) (-0.395087,-0.0139696) (-0.0918132,-0.0788735)
(-0.00994614,-0.0213973) (-0.0118322,-0.0445976) (0.00993512,0.0127006) (0.0590018,-0.387949)
(0.0139485,-0.00832193) (0.363694,-0.446652) (-0.319168,0.376483) (-0.234447,-0.0859585)
(0.173697,0.268015) (0.0279387,-0.0103741) (0.0273701,0.0937148)  (-0.055169,0.0295393)

0.244233
2.24309
3.24152
18.664

Results of MATLAB:

>>  [A,B] = eig(Rj,Rt)

A =

  Columns 1 through 3

   0.0208 - 0.0218i   0.2425 + 0.0753i  -0.1242 + 0.3753i
  -0.0234 - 0.0033i  -0.0044 + 0.0459i   0.0150 - 0.0060i
   0.0006 - 0.0162i  -0.4964 + 0.2921i   0.2719 + 0.4119i
   0.3194 + 0.0000i  -0.0298 + 0.0000i   0.0976 + 0.0000i

  Column 4

  -0.0437 - 0.1129i
   0.2351 - 0.3142i
  -0.1661 - 0.1864i
  -0.0626 + 0.0000i

B =

   0.2442         0         0         0
        0    2.2431         0         0
        0         0    3.2415         0
        0         0         0   18.6640

Eigenvalues are the same! Nice, but why Eigenvectors are not similar at all?


Answer:

There is no problem here with Eigen.

In fact for the second example run, Matlab and Eigen produced the very same result. Please remember from basic linear algebra that eigenvector are determined up to an arbitrary scaling factor. (I.e. if v is an eigenvector the same holds for alpha*v, where alpha is a non zero complex scalar.)

It is quite common that different linear algebra libraries compute different eigenvectors, but this does not mean that one of the two codes is wrong: it simply means that they choose a different scaling of the eigenvectors.

EDIT

The main problem with exactly replicating the scaling chosen by matlab is that eig(A,B) is a driver routine, which depending from the different properties of A and B may call different libraries/routines, and apply extra steps like balancing the matrices and so on. By quickly inspecting your example, I would say that in this case matlab is enforcing following condition:

but not imposing other constraints. This unfortunately means that the scaling is not unique, and probably depends on intermediate results of the generalised eigenvector algorithm used. In this case I'm not able to give you advice on how to exactly replicate matlab: knowledge of the internal working of matlab is required.

As a general remark, in linear algebra usually one does not care too much about eigenvector scaling, since this is usually completely irrelevant for the problem solved, when the eigenvectors are just used as intermediate results.

The only case in which the scaling has to be defined exactly, is when you are going to give a graphic representation of the eigenvalues.


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create vector of objects on the stack ? (c++)


c++,vector,heap-memory
I am creating a temporary vector of pointers to myObject objects. But I am wondering about what happens to the objects I created... { std::vector<myObject *> myVector; myVector.reserve(5); for (int i = 0 ; i < 5 ; ++i){ myVector[i] = new myObject(); } } I assume that at the...

How can I convert an int to a string in C++11 without using to_string or stoi?


c++,string,c++11,gcc
I know it sounds stupid, but I'm using MinGW32 on Windows7, and "to_string was not declared in this scope." It's an actual GCC Bug, and I've followed these instructions and they did not work. So, how can I convert an int to a string in C++11 without using to_string or...

How to switch Matlab plot tick labels to scientific form?


matlab,plot
I have a semilogarithmic plot which works so far with semilogx. Now I would like to change the tick labels. Now it says 10^8 10^9 ... 10^13, but I would like to see 1e8, 1e9, ... 1e13 on the x-axis. How can I change that? Cheers Manuel...

matlab plots as movie with legend


matlab,plot,legend,movie
i have a question regarding legend for movies. This is my code: fig = figure(); for i = 1: 70000 plot(signal1) hold on; plot([i,i],[-5,5]) plot(signal2,'r') hold off; title('\fontsize{14} my data'); legend('signal1','signal2'); axis tight; f(i) = getframe(fig); end The legend shows the same colors for the first two things I plot....

Why are shaders and programs stored as integers in OpenGL?


c++,opengl,opengl-es,integer,shader
I'm following the "OpenGL Superbible" book and I can't help but notice that when we create a shader and create the program that we attach the shaders to, we store them as GLuint which are unsigned integers. Why are they stored as numbers? What does the value of the number...

Plot multiple functions on one figure


matlab,matlab-figure
I'm struggling to plot multiple functions on one figure. Here is the code that I have: syms t a; a=0.9514; F1=0.5*sqrt(3*t^2); F2=-0.28375*t^2+1.155*a*(t-a)+1; F3=1; E1=diff(F1,t); E2=diff(F2,t); E3=diff(F3,t); I want to plot E1, E2 and E3, each only within a certain range, to make a "composite" line. I've tried plotting with ezplot...

MFC visual c++ LNK2019 link error


c++,mfc
I just don't understand why i can use the public variables on the class but are getting a link error when trying to use the getLicenceRefused method. I wasn't sure if the problem was because of the CString copy constructor problem I have had before so took the parameter out,...

How can I tell clang-format to follow this convention?


c++,clang-format
I would like to have this: if (!enabled) { return; } turned to this: if (!enabled) { return; } (In other words, I want short if-statements on a single line but keep the {} around them) Currently I'm using the following configuration: AllowShortIfStatementsOnASingleLine: true AllowShortLoopsOnASingleLine: true AllowShortCaseLabelsOnASingleLine: true AllowShortFunctionsOnASingleLine: true...

Interpolation inside a matrix. Matlab


matlab,matrix
I have a matrix looks like: 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 1 0 1 0 0 0 1 0 4 0 0 0 0 0 3 0 0 6 0 0 4...

3 X 3 magic square recursively


c++,algorithm,math,recursion
I'm trying to find all possible solutions to the 3X3 magic square. There should be exactly 8 solutions. My code gets them all but there are a lot of repeats. I'm having a hard time tracking the recursive steps to see why I'm getting all the repeats. // This program...

Confused about returns in stack template


c++,templates,generic-programming
I'm implementing a generic stack (with an array) in C++ and am confused about what to return in this situation: template <class T> T Stack<T>::pop(void) { if (size != 0) { return items[size - 1]; size--; } else { cerr << "Cannot pop from empty stack." << endl; return ???;...

Matlab — SVM — All Majority Class Predictions with Same Score and AUC = .50


matlab,svm,auc
I'm having a weird problem in training an SVM with an RBF kernel in Matlab. The issue is that, when doing a grid search, using 10-fold cross-validation, for the C and Sigma values I always get AUC values equal to approximately .50 (varying between .48 and .54 depending) -- I...

Incorrect Polar - Cartesian Coordinate Conversions. What does -0 Mean?


c++,polar-coordinates,cartesian-coordinates
I am getting incorrect conversions from polar to cartesian coordinates and vice versa. My code produces weird points like (1,-0). Im using this calculator to check my conversions. Also one of the conversions is completely wrong when I convert back to cartesian coordinates. Point b: (0,1) => (1,1.5708) => (0,0)...

Get an ordered list of files in a folder


c++,boost,boost-filesystem
I have used boost::filesystem::directory_iterator in order to get a list of all the available files into a given folder. The problem is that I supposed this method would give me the files in alphabetical order, while the results seem pretty random. Is there any fancy way of alphabetically sorting them?...

C++ Isn't this a useless inline declaration?


c++,inline,private,member,protected
This is another question about inlining a function. But I will take possible comments and answers right away: Defining a function inside a class makes it inline automatically. The same behaviour can be achieved by marking a function with inline outside of the class. An inline function doesn't have to...

Type function that returns a tuple of chosen types


c++,templates,c++11,metaprogramming
I've implemented a type function Tuple that turn a list of My_enum values into an std::tuple of corresponding types: #include <tuple> enum My_enum{ t_int, t_double }; // Bind_type is a type function that given a My_enum returns the corresponding type template<My_enum E> struct Bind_type; template<> struct Bind_type<t_int>{ using type =...

opencv window not refreshing at mouse callback


c++,opencv
I am trying to draw with mouse move in an opencv window. But when I draw, nothing draws on the window. When I try to close the window from the cross in the topleft(ubuntu), it opens a new window which it should be as I haven't pressed escape, and in...

undefined reference to `vtable for implementation' error


c++,build,makefile
I wrote some c++ files and after compiling with out make file it works fine . But when using make file it pop out some errors . My codes are : include directory files : application.h #ifndef APPLICATION_H #define APPLICATION_H #include "employee.h" #include "employee_data.h" #include "employee.h" ...some defintions here... #endif...

Can python script know the return value of C++ main function in the Android enviroment


python,c++
There are several ways of calling C++ executable programs. For example, we can use def run_exe_return_code(run_cmd): process=subprocess.Popen(run_cmd,stdout=subprocess.PIPE,shell=True) (output,err)=process.communicate() exit_code = process.wait() print output print err print exit_code return exit_code to process a C++ executable program: run_exe_return_code('abc') while abc is created by the following C++ codes: int main() { return 1;...