Advanced Python for Data Science Assignment 12

The following program is the familar computation of the Mandelbrot set.
```
#
# Simple Python program to calculate elements in the Mandelbrot set.
#
import numpy as np
from pylab import imshow, show

def mandel(x, y, max_iters):
    '''
    Given the real and imaginary parts of a complex number,
    determine if it is a candidate for membership in the 
    Mandelbrot set given a fixed number of iterations.
    '''
    c = complex(x, y)
    z = 0.0j
    for i in range(max_iters):
        z = z*z + c
        if (z.real*z.real + z.imag*z.imag) >= 4:
            return i
            
    return max_iters
    
def compute_mandel(min_x, max_x, min_y, max_y, image, iters):
	'''
	Calculate the mandel value for each element in the 
	image array. The real and imag variables contain a 
	value for each element of the complex space defined 
	by the X and Y boundaries (min_x, max_x) and 
	(min_y, max_y).
	'''
    height = image.shape[0]
    width = image.shape[1]
    
    pixel_size_x = (max_x - min_x) / width
    pixel_size_y = (max_y - min_y) / height
    
    for x in range(width):
        real = min_x + x * pixel_size_x
        for y in range(height):
            imag = min_y + y * pixel_size_y
            image[y, x] = mandel(real, imag, iters)
            
if __name__ == '__main__':
	image = np.zeros((1024, 1536), dtype = np.uint8)
	compute_mandel(-2.0, 1.0, -1.0, 1.0, image, 20) 
	imshow(image)
	show()
```
The program works as follows:
1. First, the image array is created with shape (1024, 1536) and initialized with zeros.
2. The compute_mandel function is called which assigns a value in the range 0 to 19 (max_iters - 1) to each element of the array.
3. The array is then plotted.
The compute_mandel function assigns a value to each element of the image array by:
1. First calculating pixel_size_x and pixel_size_y. These correspond to the smallest increments that can be represented by each element (or pixel) of the array in the ranges min_x .. max_x and min_y .. max_yrespectively.
2. Iterating over each element of the array, and computing the mandel value that corresponds to the (real, imag) value of that element. real and imag are used because you can think of the X and Y axes as the real and imaginary components of complex numbers.
The mandel function works by testing how rapidly the function z = z*z + c converges or diverges for a given value of c = complex(x,y). The value returned is the number of iterations (up to max_iters) it takes for the real and imaginary parts of z to reach the value 2 (no value greater than 2 can be part of the set). Faster divergence will result in a smaller number, slower divergence a larger number. A value within the set will return max_iters.

Try running this program to make sure you are familiar with how it works.

You are going to modify this program to work on a GPU using CUDA. You’re going to use the same mandel function, except that for it to be usable with a CUDA kernel, you need to add the @cuda.jit(device=True) decorator. This tells CUDA that mandel is a device function, which is a function that can only be executed by a kernel on the device. You also need to add the @cuda.jit decorator to the compute_mandel function as this is going to become the CUDA kernel.

Finally, you need to modify the compute_mandel function so that it can be used as a CUDA kernel. Remember that instead of iterating over every element of the array, your kernel will need to iterate over a smaller block of elements. The kernel will need to obtain the starting x and y coordinates using cuda.grid() and then calculate the ending x and y coordinates by obtaining the size of the block using gridDim and blockDim. Once you have the starting and finishing x and y coordinates, you can compute the mandel value for each element of the block as before.

The final program should look like this (use mandelbrot_gpu.py for the file name):
```
# 
# A CUDA version to calculate the Mandelbrot set
#
from numba import cuda
import numpy as np
from pylab import imshow, show

@cuda.jit(device=True)
def mandel(x, y, max_iters):
    '''
    Given the real and imaginary parts of a complex number,
    determine if it is a candidate for membership in the 
    Mandelbrot set given a fixed number of iterations.
    '''
    c = complex(x, y)
    z = 0.0j
    for i in range(max_iters):
        z = z*z + c
        if (z.real*z.real + z.imag*z.imag) >= 4:
            return i

    return max_iters

@cuda.jit
def compute_mandel(min_x, max_x, min_y, max_y, image, iters):
	'''
	YOUR COMMENT HERE
	'''
    ### YOUR CODE HERE
    
if __name__ == '__main__':
	image = np.zeros((1024, 1536), dtype = np.uint8)
	blockdim = (32, 8)
	griddim = (32, 16)
	
	image_global_mem = cuda.to_device(image)
	compute_mandel[griddim, blockdim](-2.0, 1.0, -1.0, 1.0, image_global_mem, 20) 
	image_global_mem.copy_to_host()
	imshow(image)
	show()
```
You need to replace ### YOUR CODE HERE with your code. In addition, it is essential that you replace YOUR COMMENT HERE with a comment describing how your code works.

When you are satisfied it works correctly, commit mandelbrot_gpu.py to the same repository you used in Assignment 3.

Note: don’t expect the code to run any faster unless you’re running on a real GPU.
For bonus marks, modify your previous program to use shared memory rather than global when computing the image array. Call this program mandelbrot_shared.py and commit it to the Assignment3 repository.

Programming for Data Science

Advanced Python for Data Science Assignment 12