The Quantum Physicist - 3 months ago 31

C Question

The answer to my question is most likely "No", but maybe someone has a smart solution to this problem?

Here's the problem. For example, the lapack function

`zheev()`

So my question is: Is there any way to avoid copying my matrix to a new array that stores only the lower or upper triangular part and use my current full matrix in lapack?

Answer

The example you linked on `zheev()`

**already makes use of an unpacked LDA*N=N*N matrix**. Indeed, the hermitian matrix does not need to be packed: you may not have to make a copy of your matrix. Watch out:

`zheev()`

modifies the matrix `A`

!LAPACK handles other storage mode for matrices: **see the naming scheme of LAPACK**. For instance:

`zheev()`

: the memory footprint`N*N`

and the storage are similar to the one of general unpacked`N*N`

matrices. Depending on the value of the argument`UPLO`

, the upper triangular part is used or ignored. Anyway, the matrix can be populated as if it were a general unpacked matrix of size`N*N`

. In this case, the value of the argument`UPLO`

should not change the obtained eigenvalues.`zhpev()`

: packed format. Either the upper diagonal items or the lower diagonal items are stored, depending on the value of the argument`UPLO`

. The memory footprint for matrix storage is`(N*(N+1))/2`

.`zhbev()`

: dedicated to band storage.

As you work with C or C++, here is a sample code using `zheev()`

via the interface LAPACKE. It can be compiled by `gcc main.c -o main -llapacke -llapack -lblas -lm -Wall`

. Moreover, this code ensures that **the function zheev() returns the right eigenvectors**, not the left ones. The left eigenvectors are the the complex conjugates of the right ones, as explained here.

```
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <complex.h>
#include <time.h>
#include <lapacke.h>
int main(void){
int n=200;
srand(time(NULL));
// allocate the matrix, unpacked storage
double complex** A=malloc(n*sizeof(double complex*));
if(A==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
A[0]=malloc(n*n*sizeof(double complex));
if(A[0]==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
int i;
for(i=1;i<n;i++){
A[i]=&A[0][n*i];
}
//populte the matrix, only the lower diagonal part
int j;
for(i=0;i<n;i++){
for(j=0;j<i;j++){
A[i][j]=rand()/((double)RAND_MAX)+rand()/((double)RAND_MAX)*I;
}
A[i][i]=rand()/((double)RAND_MAX)+42.;
}
//saving the matrix, because zheev() changes it.
double complex** As=malloc(n*sizeof(double complex*));
if(As==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
As[0]=malloc(n*n*sizeof(double complex));
if(As[0]==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
for(i=1;i<n;i++){
As[i]=&As[0][n*i];
}
for(i=0;i<n;i++){
for(j=0;j<i;j++){
As[i][j]=A[i][j];
}
As[i][i]=A[i][i];
}
//transpose part, conjugate
for(i=0;i<n;i++){
for(j=i+1;j<n;j++){
As[i][j]=conj(A[j][i]);
}
}
/*
for(i=0;i<n;i++){
for(j=0;j<n;j++){
printf("%g+I%g ",creal(As[i][j]),cimag(As[i][j]));
}
printf("\n");
}
*/
//let's get the eigenvalues and the eigenvectors!
double* w=malloc(n*sizeof(double));
if(w==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
int lda = n;
int info = LAPACKE_zheev(LAPACK_ROW_MAJOR, 'V', 'L', n, A[0], lda, w);
if(info<0){
fprintf(stderr,"argument %d has illegal value\n",-info);
}
if(info>0){
fprintf(stderr,"algorithm failed to converge... bad condition number ?\n");
}
//printing the eigenvalues...
for(i=0;i<n;i++){
printf("%d %g\n",i,w[i]);
}
//let's check that the column i of A is now a right eigenvectors corresponding to the eigenvalue w[i]...
int l=42;
double complex *res=malloc(n*sizeof(double complex));
if(res==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
for(i=0;i<n;i++){
res[i]=0;
for(j=0;j<n;j++){
res[i]+=As[i][j]*A[j][l];
}
res[i]-=w[l]*A[i][l];
}
double norm2res=0;
double norm2vec=0;
for(i=0;i<n;i++){
norm2res+=creal(res[i])*creal(res[i])+cimag(res[i])*cimag(res[i]);
norm2vec+=creal(A[i][l])*creal(A[i][l])+cimag(A[i][l])*cimag(A[i][l]);
}
printf("the norm of the eigenvector is %g\n",sqrt(norm2vec));
printf("||Ax-\\lambda*x||_2/||x||_2 is about %g\n",sqrt(norm2res/norm2vec));
//free the matrix
free(A[0]);
free(A);
free(As[0]);
free(As);
free(w);
free(res);
return 0;
}
```

In the code above, a copy of the matrix is performed, but that is not required by `LAPACKE_zheev()`

. Dealing with a matrix of 2000*2000, the memory footprint of the code above is about 167MB. That's more than twice of the size of the matrix (64MB) because a copy is performed. But it would be less than twice if the copy were not performed. Hence, `LAPACKE_zheev()`

does not perform any copy of the matrix. Notice also that `LAPACKE_zheev()`

allocates some space for the working array.