AMATH 483 / 583 (roche) - HW6
Due Friday May 31, 11:59pm PT
May 24, 2024
Homework 6 (80 points, 0 EC points)
1. (+20) Complex double linear system solver. Plot both the log of the residual and the log of the
normalized error ( kbAzk2
kAk1 kzk2 ✏machine ) versus the square matrix dimensions 16,**,64,...,8192 for the following
LAPACK routine. It is supported in the OpenBLAS build on Hyak. Submit your plot, and label it
accordingly.
l a p a c k i n t LAPACKE zgesv( int matrix orde r ,
l a p a c k i n t n ,
l a p a c k i n t nrhs ,
lapack compl ex doubl e ∗ a ,
l a p a c k i n t lda ,
l a p a c k i n t ∗ ipiv ,
lapack compl ex doubl e ∗ b ,
l a p a c k i n t ldb );
Use the following snippet code to initialize your matrices and rhs vectors and note the headers I use:
#include <ios t ream>
#include <complex>
#include <c s t d l i b >
#include <c s t r i n g >
#include <cmath>
#include <ve c tor>
#include <chrono>
#include <l i m it s >
#include <c b l a s . h>
#include <lapacke . h>
. . .
int main () {
. . .
a =( s td : : complex<double>∗) malloc ( s izeof ( s td : : complex<double>) ∗ ma ∗ na ) ;
b = ( s td : : complex<double>∗) malloc ( s izeof ( s td : : complex<double>) ∗ ma ) ;
z = ( s td : : complex<double>∗) malloc ( s izeof ( s td : : complex<double>) ∗ na ) ;
. . .
s rand ( 0 );
int k =0;
for ( int j = 0 ; j < na ; j++) {
for ( int i = 0 ; i < ma ; i++) {
a [ k ] = 0 . 5 − (double ) rand () / (double )RANDMAX
+ s td : : complex<double>(0 , 1)
∗ ( 0 . 5 − (double ) rand () / (double )RANDMAX) ;
i f ( i==j ) a [ k]∗= s tat ic cas t<double>(ma ) ;
k++;
}
}
s rand ( 1 );
for ( int i = 0 ; i < ma; i++) {
b [ i ] = 0 . 5 − (double ) rand () / (double )RANDMAX
+ s td : : complex<double>(0 , 1)
∗ ( 0 . 5 − (double ) rand () / (double )RANDMAX) ;
}
. . .
12. (+20) CPU-GPU data copy speed on HYAK. Write a C++ code to measure the data copy performance
between the host CPU and GPU (host to device), and between the GPU and the host CPU (device to host). Copy
8 bytes to 256MB increasing in multiples of 2. Plot the bandwidth for both directions: (bytes per second) on the
y-axis and the bu↵er size in bytes on the x-axis. Submit your plot and test code.
3. (+20) Compare FFTW to CUFFT on HYAK. Measure and plot the performance of calculating the gradient
of a 3D double complex plane wave defined on cubic lattices of dimension n3 from 163 to n = 2563, stride n⇤ = 2
for both the FFTW and CUDA FFT (CUFFT) implementations on HYAK. Let each n be measured ntrial times
and plot the average performance for each case versus n, ntrial 3. Submit your performance plot which should
have ’FLOPs’ on the y-axis (or some appropriate unit of FLOPs) and the dimension of the cubic lattices (n) on
the x-axis. You will need to estimate the operation count of computing the derivative using FFT on a lattice.
4. (+20) Fourier transforms. Evaluate the Fourier transform of the following functions by hand. Use the definitions
I provided (includes p1
2V**; , this is common in physics but also now the default used in WolframAlpha - a powerful
math AI tool) as well as the definition for Dirac delta I used in lecture if needed.
(a) f(x) = 1
p2V**; e
1
22 (xµ)2
(b) f(t) = sin(!0t) , !0 constant
(c) f(x) = ea|x| and a > 0
(d) (distribution) f(t) = (t)
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