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.
```

- 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
```

- 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)

- 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:

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:

`all(imag(V(end,:))==0)`

(the last component of each eigenvector is real)

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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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...

matlab,events,delay

In Matlab functions can be started at events,but occasionally, like with the resize function, the events are called in rapid order and the function is called many times in succession, which can cause weird behavior and lag. Is there a way to have it listen for the event but only...

c++,undefined-behavior

I was trying to save the binary equivalent of a 32 bit number in an array A. For testing my showbits() function , I choosed 8,9 when I came across this thing: I am facing an unreasonable thing in my code when I am placing memset in the function showbits(),I...

python,c++,ctypes

I have a simple test function on C++: #include <stdio.h> #include <string.h> #include <stdlib.h> #include <locale.h> #include <wchar.h> char fun() { printf( "%i", 12 ); return 'y'; } compiling: gcc -o test.so -shared -fPIC test.cpp and using it in python with ctypes: from ctypes import cdll from ctypes import c_char_p...

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...

c++,c++11,initializer-list

§[dcl.init.list] 8.5.4/2: The template std::initializer_list is not predefined; if the header <initializer_list> is not included prior to a use of std::initializer_list — even an implicit use in which the type is not named (7.1.6.4) — the program is ill-formed. Does that mean this program is ill-formed? #include <vector> int main()...

matlab,polynomial-math

I have the following code: syms z Gc=1.582*(1-0.3679*z^-1)/(1+.418*z^-1); Ghp=.3679*(z^-1)*(1+.718*z^-1)/((1-z^-1)*(1-.3679*z^-1)); T=(Gc*Ghp)/(1+Gc*Ghp); clipboard('copy', latex(simplifyFraction(T))); Which results in following for T: How can I normalise coefficients? I.e. I want the z2 in denominator and z in numerator to have the coefficient of 1. Is there any function in Matlab to do so?...

matlab,time-frequency

Code tfr = abs ( tfr ); [row_size, column_size] = size(tfr); tfr = tfr(1:round(row_size/2), 1:row_size); surf(tfr); view(2); I get in R2014b of OSX 10.10.3 Yosemite but rotating around shows that the cells should not be black Why is the output black? I wonder if this is a hardware problem or...

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....

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,...

c++

I asked a question: Detecting if an object is still active or it has been destroyed Considering that I cannot use libraries, there are no good out of the box solutions in C++. So, is it a bad practice to check if the object has been destroyed by analyzing memory...

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;...

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...