Why is Python Faster than LAPACK and C for SVD

Question

I recently had the need to compute an SVD from some C code. Given the stability and wide acceptance of LAPACK I decided to use that. The code seemed to be running a lot slower than I thought it should. I've been led to believe that Python and Numpy compute SVD via LAPACK, so I decided to compare speeds against Python. To my suprise the Python code is much faster at computing the SVD than the C code. In fact, the Python code can compute a full SVD about as fast as the C/LAPACK code computes half of an SVD.

This is my first real experience callling Fortran from C, so I suppose I am doing something wrong that is causing the slowdown, but I have no idea what it is.

Question: Why is the Python code so much faster than the C code?

In the end I need a faster SVD in C, so the follow-on question is:

Question: Is there a stable performant C library for computing partial SVD (preferrably of sparse matrices).

Here is some minimal code for timing tests. The matrix used doesn't have any particular meaning, i just needed something that I could easily generate in both C and Python.

test_svd.c

#include <stdlib.h>
#include <math.h>
#include <stdio.h>
#include <string.h>
#include <time.h>

extern void dgesvdx(
  char* jobu,
  char* jobv,
  char* range,
  int* m,
  int* n,
  double* a,
  int* lda,
  double* vl,
  double* vu,
  int* il,
  int* iu,
  int* ns,
  double* s,
  double* u,
  int* ldu,
  double* vt,
  int* ldvt,
  double* work,
  int* lwork,
  int* iwork,
  int* info);

extern void dgesdd(
  char* jobz,
  int* m,
  int* n,
  double* a,
  int* lda,
  double* s,
  double* u,
  int* ldu,
  double* vt,
  int* ldvt,
  double* work,
  int* lwork,
  int* iwork,
  int* info);

typedef enum bool {false, true} bool;

bool svd(double* a, 
         int num_rows,
         int num_cols,
         int num_sing_vals,
         double* u,
         double* s,
         double* vt,
         bool use_dgesdd) {
  double* a_work;
  int* iwork;
  double* work;
  int lwork;
  int min_m_n;
  int info = 0;
  double vl = 0;
  double vu = 1;
  int il = 1;
  int iu = num_sing_vals;
  int ns = num_sing_vals;

  //////////////////////////////////////////////////////
  // Compute and allocate optimal workspace
  //////////////////////////////////////////////////////
  min_m_n = (num_cols < num_rows) ? num_cols : num_rows;
  lwork = -1;
  // copy a to prevent corruption
  a_work = (double*)malloc(num_cols*num_rows*sizeof(double));
  // compute optimal workspace (lwork)
  double worksize;
  if(use_dgesdd) {
    iwork = (int*)malloc(7*(min_m_n)*sizeof(int));
    dgesdd("S",&num_cols,&num_rows,a_work,&num_cols,s,vt,&num_cols,u,&num_sing_vals,&worksize,&lwork,iwork,&info);
  } else {
    iwork = (int*)malloc(24*(min_m_n)*sizeof(int));
    dgesvdx("V","V","I",&num_cols,&num_rows,a_work,&num_cols,&vl,&vu,&il,&iu,&ns,s,vt,&num_cols,u,&num_sing_vals,&worksize,&lwork,iwork,&info);
  }
  if(info) {
    perror("error computing lwork\n");
    return(false);
  }
  // allocate work
  lwork = (int)worksize;
  work = (double*)malloc(lwork*sizeof(double));

  //////////////////////////////////////////////////////
  // Compute the svd
  //////////////////////////////////////////////////////
  memcpy(a_work,a,num_cols*num_rows*sizeof(double));
  if(use_dgesdd) {
    dgesdd("S",&num_cols,&num_rows,a_work,&num_cols,s,vt,&num_cols,u,&num_sing_vals,work,&lwork,iwork,&info);
  } else {
    ns = num_sing_vals;
    dgesvdx("V","V","I",&num_cols,&num_rows,a_work,&num_cols,&vl,&vu,&il,&iu,&ns,s,vt,&num_cols,u,&num_sing_vals,work,&lwork,iwork,&info);
  }
  if(info<0) {
    perror("invalid argument in SVD\n");
    return(false);
  } else if(info>0) {
    printf("SVD did not converge");
    return(false);
  }
  free(iwork);
  iwork = NULL;
  free(work);
  work = NULL;
  free(a_work);
  a_work = NULL;
  return(true);
}

int main(int argc, char** argv) {
  int num_rows = 1000;
  int num_cols = 1000;
  int size = num_rows*num_cols;
  double* A = (double*)calloc(num_rows*num_cols,sizeof(double));
  for(int r = 0; r < num_rows; ++r) {
    for(int c = 0; c < num_cols; ++c) {
      A[r*num_cols+c] = sin(5*M_PI*(r*num_cols+c)/((double)size))+cos(10*M_PI*(c*num_rows+c)/((double)size));
      if(fabs(A[r*num_cols+c])<0.75 || fabs(A[r*num_cols+c]) > 0.9) {
        A[r*num_cols+c] = 0;
      }
    }
  }
  int num_sing_vals = (num_rows < num_cols) ? num_rows : num_cols;
  double* u_sdd = (double*)calloc(num_rows*num_sing_vals,sizeof(double));
  double* s_sdd = (double*)calloc(num_sing_vals,sizeof(double));
  double* vt_sdd = (double*)calloc(num_sing_vals*num_cols,sizeof(double));
  clock_t tic = clock();
  bool success = svd(A,num_rows,num_cols,num_sing_vals,u_sdd,s_sdd,vt_sdd,true);
  clock_t toc = clock();
  printf("full dgesdd %lf seconds\n",(double)(toc-tic)/CLOCKS_PER_SEC);

  double* u_svd = (double*)calloc(num_rows*num_sing_vals,sizeof(double));
  double* s_svd = (double*)calloc(num_sing_vals,sizeof(double));
  double* vt_svd = (double*)calloc(num_sing_vals*num_cols,sizeof(double));
  tic = clock();
  success = svd(A,num_rows,num_cols,num_sing_vals,u_svd,s_svd,vt_svd,false);
  toc = clock();
  printf("full dgesvdx %lf seconds\n",(double)(toc-tic)/CLOCKS_PER_SEC);

  int first_few_sv = 200;
  double* u_partial_svd = (double*)calloc(num_rows*first_few_sv,sizeof(double));
  double* s_partial_svd = (double*)calloc(num_sing_vals,sizeof(double));
  double* vt_partial_svd = (double*)calloc(first_few_sv*num_cols,sizeof(double));
  tic = clock();
  success = svd(A,num_rows,num_cols,first_few_sv,u_partial_svd,s_partial_svd,vt_partial_svd,false);
  toc= clock();
  printf("partial dgesvdx %lf seconds\n",(double)(toc-tic)/CLOCKS_PER_SEC);

  free(A);
  free(u_sdd);
  free(s_sdd);
  free(vt_sdd);
  free(u_svd);
  free(s_svd);
  free(vt_svd);
  free(u_partial_svd);
  free(s_partial_svd);
  free(vt_partial_svd);
  return 0;
}

compiling with:

gcc test_svd.c -O3 -lm -llapack -Ddgesdd=dgesdd_ -Ddgesvdx=dgesvdx_

This gives me the following timing outputs:

full dgesdd 8.003578 seconds
full dgesvdx 17.077815 seconds
partial dgesvdx 1.498775 seconds

Comparing with the following python code:

test_svd.py

import numpy, time
dim = 1000
term1 = numpy.sin((5*numpy.pi*numpy.arange(dim**2).reshape(dim,dim))/(dim**2))
term2 = numpy.cos((10*numpy.pi*numpy.arange(dim**2).reshape(dim,dim).T)/(dim**2))
A = term1+term2
A = numpy.where(numpy.abs(A)<0.75,0,A)
A = numpy.where(numpy.abs(A)>0.9,0,A)
tic = time.time()
U,S,Vt = numpy.linalg.svd(A)
toc = time.time()
print "full svd", toc - tic, "seconds"

gives:

python2 test_svd.py
full svd 2.4460 seconds

The actual singluar values agree. However, the fact that a full SVD in Python is comparable in time to a partial SVD in C is puzzling to me.

In case it matters my python is Python 2.7.12 compiled with GCC 6.2.1 and my gcc is 6.2.1