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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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...

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...

matlab,image-processing

I have a binary matrix containing several binary objects and I want to bridge between them. Actually I have the following picture: And the result has to be like this: Is there any function or a shortcut way, other than loops, for this problem?...

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...

matlab,image-processing

I am trying to read all the images in the folder in MATLAB using this code flst=dir(str_Expfold); But it shows me output like this. which is not the sequence as i want. Can anyone please tell me how can i read all of them in sequence? for giving downmark, please...

matlab,typeconverter

I'm currently using a MATLAB to work and I need some help: I need to convert my output data (variable: units) be a double instead of a cell because I must perform a sum: units = inputdlg(question,title); sum = units + i; I've tried this code also but didn't solve...

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...

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++,file,hdf5,dataformat

We aim at using HDF5 for our data format. HDF5 has been selected because it is a hierarchical filesystem-like cross-platform data format and it supports large amounts of data. The file will contain arrays and some parameters. The question is about how to store the parameters (which are not made...

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 =...

c++

As a continuation of a: Thread, I came across a problem with writing a method of a class which returns: std::vector<std::unique_ptr<Object>> I get compiler errors when such a return type is written. There is some problem with delete operand or something ... Generally, I've wanted to write a method which...

c++

I define a template class in which, I define two type-cast operator template <class base_t> struct subclass { base_t base; //any function which defined for 'base_t' can be used with 'subclass<base_t>' operator base_t&() { return base; } //I want 'subclass<base_t>' can be converted to any class which 'base_t' can //I...

c++,templates

I'm trying to understand C++ template templates by implementing a generic container class. Here is the code: using namespace std; template <typename T, template <typename STORETYPE> class Container> class Store { public: ~Store() {}; Store() {}; void someFunc( const T & ) {}; //... private: Container<T> storage; }; int main(int...

c++,arrays,string

I was trying to achieve translating a character array into a integer string and corresponding character to their alphabetical order. For instance: A(a) = 0 , Z(z) = 25. string key_char = argv[1]; string key_num; for (int i = 0; i < key_char.length(); i++){ if (isalpha(key_char[i])){ if (islower(key_char[i])){ key_num[i] =...