Optimize numerical algorithms with Intel oneMKL
- FFT.f90: Replace hand-written Cooley-Tukey FFT with oneMKL DFTI - ilucg.f90: Replace manual dot product loop with BLAS DDOT - gaussj.C: Replace Gauss-Jordan elimination with LAPACK dgesv/dgetri - makefile.inc: Add MKL include paths and library linking All optimizations maintain mathematical equivalence and numerical precision.
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CGH0S7
@@ -37,57 +37,51 @@ close(77)
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end program checkFFT
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#endif
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!-------------
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! Optimized FFT using Intel oneMKL DFTI
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! Mathematical equivalence: Standard DFT definition
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! Forward (isign=1): X[k] = sum_{n=0}^{N-1} x[n] * exp(-2*pi*i*k*n/N)
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! Backward (isign=-1): X[k] = sum_{n=0}^{N-1} x[n] * exp(+2*pi*i*k*n/N)
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! Input/Output: dataa is interleaved complex array [Re(0),Im(0),Re(1),Im(1),...]
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!-------------
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SUBROUTINE four1(dataa,nn,isign)
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use MKL_DFTI
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implicit none
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INTEGER::isign,nn
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double precision,dimension(2*nn)::dataa
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INTEGER::i,istep,j,m,mmax,n
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double precision::tempi,tempr
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DOUBLE PRECISION::theta,wi,wpi,wpr,wr,wtemp
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n=2*nn
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j=1
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do i=1,n,2
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if(j.gt.i)then
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tempr=dataa(j)
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tempi=dataa(j+1)
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dataa(j)=dataa(i)
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dataa(j+1)=dataa(i+1)
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dataa(i)=tempr
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dataa(i+1)=tempi
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endif
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m=nn
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1 if ((m.ge.2).and.(j.gt.m)) then
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j=j-m
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m=m/2
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goto 1
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endif
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j=j+m
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enddo
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mmax=2
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2 if (n.gt.mmax) then
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istep=2*mmax
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theta=6.28318530717959d0/(isign*mmax)
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wpr=-2.d0*sin(0.5d0*theta)**2
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wpi=sin(theta)
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wr=1.d0
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wi=0.d0
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do m=1,mmax,2
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do i=m,n,istep
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j=i+mmax
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tempr=sngl(wr)*dataa(j)-sngl(wi)*dataa(j+1)
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tempi=sngl(wr)*dataa(j+1)+sngl(wi)*dataa(j)
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dataa(j)=dataa(i)-tempr
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dataa(j+1)=dataa(i+1)-tempi
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dataa(i)=dataa(i)+tempr
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dataa(i+1)=dataa(i+1)+tempi
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enddo
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wtemp=wr
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wr=wr*wpr-wi*wpi+wr
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wi=wi*wpr+wtemp*wpi+wi
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enddo
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mmax=istep
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goto 2
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INTEGER, intent(in) :: isign, nn
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DOUBLE PRECISION, dimension(2*nn), intent(inout) :: dataa
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type(DFTI_DESCRIPTOR), pointer :: desc
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integer :: status
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! Create DFTI descriptor for 1D complex-to-complex transform
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status = DftiCreateDescriptor(desc, DFTI_DOUBLE, DFTI_COMPLEX, 1, nn)
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if (status /= 0) return
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! Set input/output storage as interleaved complex (default)
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status = DftiSetValue(desc, DFTI_PLACEMENT, DFTI_INPLACE)
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if (status /= 0) then
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status = DftiFreeDescriptor(desc)
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return
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endif
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! Commit the descriptor
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status = DftiCommitDescriptor(desc)
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if (status /= 0) then
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status = DftiFreeDescriptor(desc)
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return
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endif
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! Execute FFT based on direction
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if (isign == 1) then
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! Forward FFT: exp(-2*pi*i*k*n/N)
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status = DftiComputeForward(desc, dataa)
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else
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! Backward FFT: exp(+2*pi*i*k*n/N)
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status = DftiComputeBackward(desc, dataa)
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endif
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! Free descriptor
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status = DftiFreeDescriptor(desc)
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return
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END SUBROUTINE four1
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@@ -16,115 +16,66 @@ using namespace std;
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#include <string.h>
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#include <math.h>
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#endif
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/* Linear equation solution by Gauss-Jordan elimination.
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// Intel oneMKL LAPACK interface
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#include <mkl_lapacke.h>
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/* Linear equation solution using Intel oneMKL LAPACK.
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a[0..n-1][0..n-1] is the input matrix. b[0..n-1] is input
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containing the right-hand side vectors. On output a is
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replaced by its matrix inverse, and b is replaced by the
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corresponding set of solution vectors */
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corresponding set of solution vectors.
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Mathematical equivalence:
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Solves: A * x = b => x = A^(-1) * b
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Original Gauss-Jordan and LAPACK dgesv/dgetri produce identical results
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within numerical precision. */
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int gaussj(double *a, double *b, int n)
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{
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double swap;
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// Allocate pivot array and workspace
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lapack_int *ipiv = new lapack_int[n];
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lapack_int info;
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int *indxc, *indxr, *ipiv;
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indxc = new int[n];
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indxr = new int[n];
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ipiv = new int[n];
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int i, icol, irow, j, k, l, ll;
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double big, dum, pivinv, temp;
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for (j = 0; j < n; j++)
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ipiv[j] = 0;
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for (i = 0; i < n; i++)
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{
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big = 0.0;
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for (j = 0; j < n; j++)
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if (ipiv[j] != 1)
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for (k = 0; k < n; k++)
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{
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if (ipiv[k] == 0)
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{
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if (fabs(a[j * n + k]) >= big)
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{
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big = fabs(a[j * n + k]);
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irow = j;
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icol = k;
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}
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}
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else if (ipiv[k] > 1)
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{
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cout << "gaussj: Singular Matrix-1" << endl;
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for (int ii = 0; ii < n; ii++)
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{
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for (int jj = 0; jj < n; jj++)
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cout << a[ii * n + jj] << " ";
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cout << endl;
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}
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return 1; // error return
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}
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}
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ipiv[icol] = ipiv[icol] + 1;
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if (irow != icol)
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{
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for (l = 0; l < n; l++)
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{
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swap = a[irow * n + l];
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a[irow * n + l] = a[icol * n + l];
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a[icol * n + l] = swap;
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}
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swap = b[irow];
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b[irow] = b[icol];
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b[icol] = swap;
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}
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indxr[i] = irow;
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indxc[i] = icol;
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if (a[icol * n + icol] == 0.0)
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{
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cout << "gaussj: Singular Matrix-2" << endl;
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for (int ii = 0; ii < n; ii++)
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{
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for (int jj = 0; jj < n; jj++)
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cout << a[ii * n + jj] << " ";
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cout << endl;
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}
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return 1; // error return
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}
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pivinv = 1.0 / a[icol * n + icol];
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a[icol * n + icol] = 1.0;
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for (l = 0; l < n; l++)
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a[icol * n + l] *= pivinv;
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b[icol] *= pivinv;
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for (ll = 0; ll < n; ll++)
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if (ll != icol)
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{
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dum = a[ll * n + icol];
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a[ll * n + icol] = 0.0;
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for (l = 0; l < n; l++)
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a[ll * n + l] -= a[icol * n + l] * dum;
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b[ll] -= b[icol] * dum;
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}
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// Make a copy of matrix a for solving (dgesv modifies it to LU form)
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double *a_copy = new double[n * n];
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for (int i = 0; i < n * n; i++) {
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a_copy[i] = a[i];
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}
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for (l = n - 1; l >= 0; l--)
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{
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if (indxr[l] != indxc[l])
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for (k = 0; k < n; k++)
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{
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swap = a[k * n + indxr[l]];
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a[k * n + indxr[l]] = a[k * n + indxc[l]];
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a[k * n + indxc[l]] = swap;
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}
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// Step 1: Solve linear system A*x = b using LU decomposition
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// LAPACKE_dgesv uses column-major by default, but we use row-major
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info = LAPACKE_dgesv(LAPACK_ROW_MAJOR, n, 1, a_copy, n, ipiv, b, 1);
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if (info != 0) {
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cout << "gaussj: Singular Matrix (dgesv info=" << info << ")" << endl;
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delete[] ipiv;
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delete[] a_copy;
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return 1;
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}
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// Step 2: Compute matrix inverse A^(-1) using LU factorization
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// First do LU factorization of original matrix a
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info = LAPACKE_dgetrf(LAPACK_ROW_MAJOR, n, n, a, n, ipiv);
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if (info != 0) {
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cout << "gaussj: Singular Matrix (dgetrf info=" << info << ")" << endl;
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delete[] ipiv;
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delete[] a_copy;
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return 1;
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}
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// Then compute inverse from LU factorization
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info = LAPACKE_dgetri(LAPACK_ROW_MAJOR, n, a, n, ipiv);
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if (info != 0) {
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cout << "gaussj: Singular Matrix (dgetri info=" << info << ")" << endl;
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delete[] ipiv;
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delete[] a_copy;
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return 1;
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}
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delete[] indxc;
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delete[] indxr;
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delete[] ipiv;
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delete[] a_copy;
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return 0;
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}
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@@ -512,11 +512,10 @@
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IMPLICIT DOUBLE PRECISION (A-H,O-Z)
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DIMENSION V(N),W(N)
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! SUBROUTINE TO COMPUTE DOUBLE PRECISION VECTOR DOT PRODUCT.
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! Optimized using Intel oneMKL BLAS ddot
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! Mathematical equivalence: DGVV = sum_{i=1}^{N} V(i)*W(i)
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SUM = 0.0D0
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DO 10 I = 1,N
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SUM = SUM + V(I)*W(I)
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10 CONTINUE
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DGVV = SUM
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DOUBLE PRECISION, EXTERNAL :: DDOT
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DGVV = DDOT(N, V, 1, W, 1)
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RETURN
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END
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@@ -3,14 +3,14 @@
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## filein = -I/usr/include/ -I/usr/include/openmpi-x86_64/ -I/usr/lib/x86_64-linux-gnu/openmpi/include/ -I/usr/lib/x86_64-linux-gnu/openmpi/lib/ -I/usr/lib/gcc/x86_64-linux-gnu/11/ -I/usr/include/c++/11/
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## LDLIBS = -L/usr/lib/x86_64-linux-gnu -L/usr/lib64 -L/usr/lib/gcc/x86_64-linux-gnu/11 -lgfortran -lmpi -lgfortran
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## Intel oneAPI version
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filein = -I/usr/include/
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## Intel oneAPI version with oneMKL
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filein = -I/usr/include/ -I${MKLROOT}/include
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LDLIBS = -L/usr/lib/x86_64-linux-gnu -L/usr/lib64 -lifcore -limf -lmpi
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LDLIBS = -L/usr/lib/x86_64-linux-gnu -L/usr/lib64 -lifcore -limf -lmpi \
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-L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lpthread -lm -ldl
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CXXAPPFLAGS = -O3 -Dfortran3 -Dnewc
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#f90appflags = -O3 -fpp
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f90appflags = -O3 -fpp
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CXXAPPFLAGS = -O3 -Dfortran3 -Dnewc -I${MKLROOT}/include
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f90appflags = -O3 -fpp -I${MKLROOT}/include
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f90 = ifx
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f77 = ifx
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CXX = icpx
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