# Project 4: Numc

## Overview

Due Sunday, May 3rd

This project is designed to be both a C project as well as a performance project. In this project you will be implementing a slower version of numpy. Your version of numpy, numc (how exciting!), is most likely to be slower than numpy, but much faster than the naive implementations of matrix operations. You will first complete a naive solution for some matrix functions in C, then you will experiment with the setup file in Python to install your numc module. After that, you will gain a deeper understanding of the Python-C interface by overloading some operators and defining some instance methods for numc.Matrix objects. Finally, you will speed up your naive solution, thus making numc.Matrix operations faster.

Do not expect your final completed numc module to be as good as numpy, but you should expect a very large speedup compared to the naive solution, especially for matrix multiplication and exponentiation!

## Tips and Guidelines

• Please start early! Because there are many more 61C students than Hive machines, you will likely share resources with your classmates. This might affect the measurement of your code’s speedup. We encourage you to use Hivemind to help balance the load amongst the hive machines.
• You can either allocate new matrices and throw errors in numc.c or matrix.c, but you have to make sure you throw a type error if the inputs are not valid, an index error if the indices you are trying to access are out of range, or a runtime error if any matrix allocation fails during the execution of a number method
• We will not be directly testing your C code. All tests will be in Python!
• You may change the function signatures in matrix.h and matrix.c as you deem appropriate, but you may not change the function signatures in numc.h and numc.c.
• You may change the skeleton code in numc.c, especially if you are not using a row-major setup for your matrices.
• If you would like to run the reference solution to compare your solution to, you can import the dumbpy library on hive as we have already installed it there for you!
• For this project, we strongly suggest working on Hive machines under you cs61c account. We have set up a few environment settings that only work if you use your cs61c account.

## Getting Started

Ssh into one of the hive machines under your cs61c class account.

Please follow the directions in this Google Form to get a repository: https://forms.gle/ZLgGJhggiDvBja667. After completing the form, clone your GitHub Classroom repository and add the starter code repository as a remote:

$git clone YOUR_REPO_NAME$ cd YOUR_REPO_NAME
$git remote add starter https://github.com/61c-teach/sp20-proj4-starter.git  If we publish changes to the starter code, retrieve them using git pull starter master. To be able install the modules that you will complete in this project, you must create a virtual environment with by running $ conda create -y -n proj4env python=3.6


Respond to the prompt to install packages with “y” (with no quotes). You can (and should) ignore any warnings about conda being out of date. These will take about 30 seconds to install. Finally, run the following command to activate the virtual environment:

$conda activate proj4env  This will put you in a virtual environment needed for this project. Please remember that if you exit the virtual environment and want to return to work on the project, you must re-run conda activate proj4env. This also means every time you re-ssh into the hive, you will have to re-run conda activate proj4env. Finally, if you have to exit out of the virtual environment, you can do so by running: $ conda deactivate


We already have the reference library dumbpy installed for you on Hive machines. You can import it with or without the virtual environment, and all object and function names are the same as the numc module that you will implement (please refer to Task 3). You will only be able to access the dumbpy package on hive as we will not be directly releasing it. You can use it as a reference for both correctness and speed.

Again, for this project, we strongly suggest working on Hive machines in your cs61c class account. You will not be able to call conda or import dumbpy if you are using other class accounts. We may be unable to help you with issues caused by working outside of the Hive.

## Task 1: Matrix functions in C

There has been an important update since when we first released the project. Please read this linked Piazza post carefully!!

For this task, you will need to complete all functions in matrix.c labelled with /* TODO: YOUR CODE HERE */. The comments above each function signature in matrix.c contain instructions on how to implement the functions, so read them carefully before you start coding.

The matrix struct is defined in matrix.h. Feel free to change it, but make sure your changes are compatible with our starter code.

typedef struct matrix {
int rows;
int cols;
double* data;
int ref_cnt;
struct matrix *parent;
} matrix;


rows is the number of rows of this matrix, cols is the number of columns, and data is a 1D representation of the 2D matrix data. ref_cnt is the number of existing matrix structs (including itself) that share all or part of the data array with this particular matrix struct. parent indicates whether this matrix struct is a slice of another matrix, and should be set to its parent matrix struct if it is and NULL otherwise.

matrix.h also imports the library Python.h, but for this part you should not need any other functions besides PyErr_SetString.

Depending on your implementation of matrix.c and numc.c, you may or may not assume that result is already preallocated or that all inputs’ dimensions are valid. However, as mentioned in the Tips and Guidelines section, your Python number methods will need to handle the case where matrix allocation fails.

The function allocate_matrix_ref is called from numc.c’s Matrix61c_subscript function and and is used for getting a row of the from matrix (see Info: numc.Matrix indexing for an example). Currently, Matrix61c_subscript and allocate_matrix_ref assume a row-major setup. If you choose to implement your matrices as column-major, you will have to change the implementation of Matrix61c_subscript, and you might also want to change the function signature of allocate_matrix_ref.

Again, you may change any function signature in matrix.h and matrix.c.

### Testing for Correctness

We’ve provided some sanity in mat_test.c. These tests make several assumptions:

• They assume that all result matrices are already pre-allocated with the correct dimensions and that all input dimensions are valid.
• They assume that you have not modified the matrix struct in matrix.h
• All tests except the tests for get and set assume that your get and set are correct
• They assume that you have not modified the function signatures.

Violation of one or more of these assumptions may not cause your tests to fail, but please keep this in mind if your tests are failing and you are violating at least one of these assumptions.

To run the CUnit tests, run

$make test  in the root folder of your project. This will create an executable called test in the root folder and run it. By default, CUnit will run these tests in Normal mode. When debugging a specific issue, it may be helpful to switch to Verbose mode, which can be done by commenting and uncommenting the relevant lines in mat_test.c: // CU_basic_set_mode(CU_BRM_NORMAL); CU_basic_set_mode(CU_BRM_VERBOSE);  Make sure that one line is uncommented at a time. Please keep in mind that these tests are not comprehensive, and passing all the sanity tests does not necessarily mean your implementation is correct. This is especially true with the memory functions allocate_matrix, allocate_matrix_ref, and deallocate_matrix. Also keep in mind that the autograder will be using our own set of sanity tests, and will not be running your CUnit tests. Another thing to note is that the Makefile is written for compilation on the hive machines. If you wish to run it locally, you will have to modify the Makefile by replacing the path to your CUnit/Python libraries in your CUNIT and PYTHON variables. You will also need to make sure that your local computer supports AVX extensions and OpenMP. Finally, you are welcomed to modify the mat_test.c file in the tests directory to implement your custom test cases. ## Task 2: Writing the setup file The setup.py file is used for installing your custom-built modules. After completing it, you should be able to install numc by simply running: $ make


This will uninstall your previously installed numc module if it existed and reinstall numc. We have written numc.c so that numc.Matrix will be initialized and ready to import upon succesful installation of the numc module. You should rerun make every time you make changes and want them to be reflected in the numc module.

You can uninstall your numc module by running

$make uninstall  We have provided you with the compiler and linker flags in setup.py, and your task is to find out how to use them to build your module. You will likely get a lot of warnings about functions being defined but not used, and that’s ok! You should ignore these warnings for now, and they will be gone after you finish writing Task 3. Remember that you must be in the virtual environment that you set up in order to install the modules, otherwise you will get a “Read-only file system” error. READ FIRST: take a look at the function distutils.core.setup (https://docs.python.org/3.6/distutils/apiref.html), and here is an example usage. You only need two function calls to complete this section, if you’re doing more than that, please reread the docs included as you’re likely doing something wrong ## Task 3: Writing the Python-C interface Now that you have successfully installed your numc module, you can import your numc.Matrix objects in Python programs! Here are some ready-to-use features already implemented for numc.Matrix objects. You might find them helpful when debugging Task 3. ### Info: Importing numc.Matrix Here are several ways of importing numc.Matrix from numc import Matrix import numc numc.Matrix import numc as nc nc.Matrix  ### Info: numc.Matrix initialization The code block below lists all the different ways of creating a numc.Matrix object. import numc as nc # This creates a 3 * 3 matrix with entries all zeros mat1 = nc.Matrix(3, 3) # This creates a 2 * 3 matrix with entries all ones mat2 = nc.Matrix(3, 3, 1) # This creates a 2 * 3 matrix with first row 1, 2, 3, second row 4, 5, 6 mat3 = nc.Matrix([[1, 2, 3], [4, 5, 6]]) # This creates a 1 * 2 matrix with entries 4, 5 mat4 = nc.Matrix(1, 2, [4, 5])  More specifically: • nc.Matrix(rows: int, cols: int) will create a matrix with rows rows and cols cols. All entries in this matrix are defaulted to 0. • nc.Matrix(rows: int, cols: int, val: float) will create a matrix with rows rows and cols cols. All entries in this matrix will be initialized to val. • nc.Matrix(rows: int, cols: int, lst: List[int]) will create a matrix with rows rows and cols cols. lst must have length rows * cols, and entries of the matrix will be initialized to values of lst in a row-major order. • nc.Matrix(lst: List[List[int]]) will create a matrix with the same shape as the 2D lst (i.e. each list in lst is a row for this matrix). ### Info: numc.Matrix indexing You can index into a matrix and change either the value of one single entry or an entire row. More specifically, mat[i] should give you the ith row of matrix. If mat has more than 1 column, mat[i] should also be of type numc.Matrix with (mat’s number of columns, 1) as its shape. In other words, mat[i] returns a column vector. This is to support 2D indexing of numc matrices. If mat only has one column, then mat[i] will return a double. mat[i][j] should give you the entry at the ith row and jth column. If you are setting one single entry by indexing, the data type must be float or int. If you are setting an entire row of a matrix that has more than one column, you must provide a 1D list that has the same length as the number of columns of that matrix. Every element of this list must be either of type float or int. Please note that if mat[i] has more than 1 entry, it will share data with mat, and changing mat[i] will result in a change in mat. The example given below assumes the matrices are initialized from the code block above. # mat1 print(mat1[0]) # this gives the 0th row of mat1, should print out [[0.0], [0.0], [0.0]] print(mat1[0][1]) # this should print out 0 mat1[0][1] = 5 print(mat1) # this should print out [[0.0, 5.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]] mat1[0] = [4, 5, 6] print(mat1) # this should print out [[4.0, 5.0, 6.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]] # mat2 print(mat2) # [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0], [1.0, 1.0, 1.0]] mat2[1][1] = 2 print(mat2[1]) # [[1.0], [2.0], [1.0]] # You can change a value in a slice, and that will change the original matrix print(mat2) # [[1.0, 1.0, 1.0], [1.0, 2.0, 1.0], [1.0, 1.0, 1.0]] mat2_slice = mat2[0] # [[1.0], [1.0], [1.0]] mat2_slice[0] = 5 print(mat2_slice) # [[5.0], [1.0], [1.0]] print(mat2) # [[5.0, 1.0, 1.0], [1.0, 2.0, 1.0], [1.0, 1.0, 1.0]]  Partial slices, however, are not supported. For example, mat[1:3] # not allowed mat[0][1:3] # not allowed  ### Info: instance attributes The matrices and vectors have an attribute shape, which is a tuple of (rows, cols). Example is given below. print(mat1.shape) # this should print out (3, 3)  ### Info: Python/C API Reference Here is the link to the full reference manual: https://docs.python.org/3.6/c-api/index.html. If you ever find anything confusing in the skeleton code or are at a lost on how to implement numc.c, this is a great resource. ### Quick Overview of numc skeleton code We define the Matrix61c struct in numc.h. It is of type PyObject (this means you can always cast Matrix61c to PyObject, but not vice versa), which according to the official documentation, “contains the information Python needs to treat a pointer to an object as an object”. Our Matrix61c has the matrix struct we defined in matrix.h. Then we define a struct PyTypeObject named Matrix61cType to specify the intended behaviors of our Python object Matrix61c. This struct will then be initialized to be our numc.Matrix objects. static PyTypeObject Matrix61cType = { PyVarObject_HEAD_INIT(NULL, 0) .tp_name = "numc.Matrix", .tp_basicsize = sizeof(Matrix61c), .tp_dealloc = (destructor)Matrix61c_dealloc, .tp_repr = (reprfunc)Matrix61c_repr, .tp_as_number = &Matrix61c_as_number, .tp_flags = Py_TPFLAGS_DEFAULT | Py_TPFLAGS_BASETYPE, .tp_doc = "numc.Matrix objects", .tp_methods = Matrix61c_methods, .tp_members = Matrix61c_members, .tp_as_mapping = &Matrix61c_mapping, .tp_init = (initproc)Matrix61c_init, .tp_new = Matrix61c_new };  For example, .tp_dealloc tells Python which function to call to destroy a numc.Matrix object when its reference count becomes 0, and .tp_members tells Python what instance attributes numc.Matrix objects have. You can take a look at the official documentation if you are curious. ### Useful functions: Here is a list of some functions and Python objects from <Python.h> that you may find useful. You can also choose any other functions at this link. Now you are ready to complete numc.c, the Python-C interface! As before, you will need to fill out all functions and variables labeled /* TODO: YOUR CODE HERE */. The code for initializing the module numc and the object type numc.Matrix is already done for you. Although not required, we encourage you to take a look at the existing code to better understand the interface. Below are the two main parts for this task. ### Number Methods For this part, we ask you to overload operators for numc.Matrix objects. Here are the expected behaviors of overloaded operators: • a + b: Element-wise sum of a and b. a and b must have the same dimensions. Returns a numc.Matrix object. You will have to throw a type error if they don’t. • a - b: Element-wise subtraction of a and b. a and b must have the same dimensions. Returns a numc.Matrix object. You will have to throw a type error if they don’t. • a * b: Matrix multiplication of a and b. a’s number of columns must be equal to b’s number of rows. Returns a numc.Matrix object. You will have to check for those conditions and throw a type error if arguments are invalid. Remember that this is a matrix multiplication, not an element-wise multiplication. • -a: Element-wise negation of a. Returns a numc.Matrix object. • abs(a): Element-wise absolute value of a. Returns a numc.Matrix object. • a ** pow: Raise a to the powth power. pow must be a positive integer and a must be a square matrix. Returns a numc.Matrix object. You will have throw a type error if matrix is not square or if pow is non-positive. This operator is defined in terms of matrix multiplication, not element-wise multiplication. Please note that for all these operations above, you never directly modify the matrix that you pass in. You always make a new numc.Matrix object to hold your result, so make sure you set the shape attribute of the new numc.Matrix. You can use Matrix61c_new to create new numc.Matrix objects. Take a look at the implementation of Matrix61c_subscript for an example. For all the functions above, throw a runtime error if any error occurs (such as matrix allocation failure) and causes the operation to fail. Moreover, for any operations that involve two instances of numc.Matrix, you will have to make sure that the second instance b is indeed of type numc.Matrix as we do not support operations between numc.Matrix and other data/object types. Please read the comments in numc.c carefully. Here is a table that tells you which function in numc.c in which you will implement each of the above operators Operator Function + Matrix61c_add - (subtraction) Matrix61c_sub * Matrix61c_multiply - (negation) Matrix61c_neg abs() Matrix61c_abs ** Matrix61c_pow All these functions will be called through a Python-C interface after you complete the numc module. In other words, these are the functions that will be called when you do matrix operations with numc.Matrix objects, and these interface methods will call matrix.c methods that you just implemented. You will have to check for the validity of the dimensions before actually carrying out the arithmetic, and throw an error if needed. Specifically, throw an type error if the arguments’ dimensions are invalid, and a runtime error if any memory allocation fails during execution. Again, depending on your implementation, these error checks could either be in matrix.c or numc.c After you implement all the functions above, you will need to fill out the struct Matrix61c_as_number, which is used to define the object type numc.Matrix. Here is the link to the official documentation of a PyNumberMethods struct: https://docs.python.org/3/c-api/typeobj.html#c.PyNumberMethods ### Instance Methods You will implement two instance methods for numc.Matrix: • set(self: numc.Matrix, i: int, j: int, val: float): Set self’s entry at the ith row and jth column to val. Throw a type error if the number of arguments parsed from args is not 3 or if the arguments are of the wrong types. Throw an index error if either i, j, or both are out of range. Return None. • get(self: numc.Matrix, i: int, j: int): Returns the entry at the ith row and jth column. Throw a type error if the number of arguments parsed from args is not 2 or if the arguments are of the wrong types. Throw an index error if either i, j, or both are out of range. Return value is a Python float. These functions will call get and set in matrix.c to actually get or set the value. Again, you can throw errors either in numc.c or matrix.c. Here is a table that tells you which functions in numc.c in which you will implement each of the above instance methods Python method C Function set Matrix61c_set_value get Matrix61c_get_value After you implement all the functions above, you will need to fill out the array of PyMethodDef structs Matrix61c_methods, which is used to define the object type numc.Matrix. This link tells you what goes into a PyMethodDef struct: https://docs.python.org/3/c-api/structures.html ### Indexing As mentioned in task 1, if you are storing your matrix data in a non-row-major order, you might want to change your Matrix61c_subscript. Regardless of how you are storing you matrices, now is a good time to check if your allocate_matrix_ref in matrix.c is working as intended. A correct implementation of allocate_matrix_ref and Matrix61c_subscript should result in behaviors specified in the indexing info section above. ### How to Debug To debug the Python-C interface, we suggest that you write your test files in Python, and use gdb or both gdb and pdb to debug. #### Using only gdb You don’t have to use pdb if you do not wish to set breakpoints in your Python test file. Open your terminal and run $ gdb python3


Then you can set breakpoints in your C files using the normal gdb commands. gdb will warn you with

No source file named {your c file}
Make breakpoint pending on future shared library load? (y or [n])


Press ‘y’ (without the quotes) to proceed.

After that, you can run run {your python test file name}.py in gdb, and gdb will break at the breakpoints that you just set.

#### Using both gdb and pdb

You will have to use pdb if you wish to set breakpoints in your Python file. Here’s how it works. Start gdb by running

\$ gdb python3


and set your breakpoints in C (see previous section). Then you will need to run in gdb

run -m pdb {your python file}.py


After this step, you can set breakpoints in your Python file using gdb syntax (for example, b test.py:5). With this approach, your debugger will switch between pdb and gdb depending on whether you are stepping through a Python file or a C file.

## Task 4: Speeding up matrix operations

Now that you have completed the three steps above and successfully installed your naive version of numc, it’s time to speed up your matrix functions in matrix.c! Below we outline some steps for boosting performance.

### Step 1: Unrolling and Other Optimizations

You should first try to speed up the computation by trying to apply conventional code optimizations (i.e. without using SSE or OpenMP). While we won’t tell you the exact steps, here are some hints that should help you get started:

1. Function calls are expensive since they involve setting up a stack frame and jumping to a different part of code. See if there are any functions that are frequently called that don’t necessarily need to be.
2. Are there any places where you could do manual loop unrolling?
3. Is there any unnecessary computation being done?

Note that the above hints relate to general optimization practices. You do not necessarily need to do all of these to achieve a good speedup.

Once you have improved performance using these optimizations, you can start applying vectorization and parallelization to make the program even faster. Note that you have considerable freedom to apply any of these optimizations, and there is more than one correct solution. Try to experiment with different approaches and see which one gives you the best performance.

### Step 2: SIMD Instructions

From lectures, you learned how to apply SIMD instructions to improve performance. The processors in the hive machines support the Intel AVX extensions, which allow you to do SIMD operations on 256 bit values (not just 128 bit, as we have seen in the lab). You should use these extensions to perform four operations in parallel (since all floating point numbers are doubles, which are 64 bit in size). If you are unfamiliar with SIMD instructions, lab 12 can be a good warmup.

As a reminder, you can use the Intel Intrinsics Guide as a reference to look up the relevant instructions. You will have to use the __m256d type to hold 4 doubles in a YMM register, and then use the _mm256_* intrinsics to operate on them.

Here is a list of AVX instructions that you may find helpful, although you are also allowed to use other AVX instructions not on the list.

void _mm256_storeu_pd (double * mem_addr, __m256d a)
__m256d _mm256_set1_pd (double a)
__m256d _mm256_set_pd (double e3, double e2, double e1, double e0)
__m256d _mm256_add_pd (__m256d a, __m256d b)
__m256d _mm256_sub_pd (__m256d a, __m256d b)
__m256d _mm256_fmadd_pd (__m256d a, __m256d b, __m256d c)
__m256d _mm256_mul_pd (__m256d a, __m256d b)
__m256d _mm256_cmp_pd (__m256d a, __m256d b, const int imm8)
__m256d _mm256_and_pd (__m256d a, __m256d b)
__m256d _mm256_max_pd (__m256d a, __m256d b)


### Step 3: OpenMP

Finally you should use OpenMP to parallelize computation. Note that you will need to make sure that none of the different threads overwrites each others’ data. Just adding a #pragma omp parallel for may cause errors.

Note that the Hive machines have 4 cores with two hyperthreads each. This means that you should expect a speed-up of 4-8x (note that hyperthreads mean that two different threads execute on the same physical core at the same time; they will therefore compete for processor resources, and as a result, you will not get the same performance as if you were running on two completely separate cores).

### Testing Speedup

As mentioned in Tips and Guidelines and Getting Started, We have installed the naive solution which we will be comparing against on hive! The python package is called dumbpy and you can import it like any other python library (so long as you are on hive)! Please note we will not be distributing this binary which means you must work on hive if you want to test with it. You should use this and the time package to determine how much you sped up your code.

## Task 5: Tell us what you did!

Write up what you did in your README.md! While we do not have a specific format we are looking for, you should discuss what you did as a whole, the different python functions you implemented, what performance improvements you had, what were you surprised about, etc. It should be at least a page long (aka more than both 500 words/3000 characters). You will be graded on this so make sure you do it or you will lose points!

The grading breakdown for Project 4 is as follows:

• Correctness: 55%
• Speedup: 40%
• Multiplication: 20%
• Power: 25%
• Simple: 12.5%
• Comprehensive: 42.5%

Here are the graphs for the speedup tests. The x-axis is your speedup times and the y-axis indicates what percentage of that test’s total score you will receive.

Multiplication Power
Simple Comprehensive

Minimum speedup for 100% on each test:

• Multiplication: 95x
• Power: 1900x
• Simple: 5x
• Comprehensive: 98x

Since we are running your submissions on hive, albeit reserved, speedup times may fluctuate a bit. You should try to go above the speedup value as we will rerun the ag after the deadline and your speedup may go up or down. We will not rerun submissions if they went down unless we made another change to the autograder.

We will only be using your matrix.h, matrix.c, numc.c, setup.py, and README.md for grading. Gradescope shows you all of the correctness tests we have currently written. The tests you can see on gradescope are trying to be comprehensive though we may add more tests if we find this test suit is missing some tests, so be aware tests may be added even after the deadline. We do not know how many more we will add as we are looking at common problems to see which tests would be good to add.