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Numerical Software Development

Numerical Software Development Course Notes

Copyright 2021, Yung-Yu Chen All rights reserved.

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Objectives: This course discusses the art to build numerical software, i.e., computer programs applying numerical methods for solving mathematical or physical problems. We will be using the combination of Python and C++ and related tools (e.g., bash, git, make, etc.) to learn the modern development processes. By completing this course, students will acquire the fundamental skills for developing modern numerical software.

Prerequisites: This is a graduate or senior level course open to students who have taken computer architecture, engineering mathematics or equivalents. Working knowledge of Linux and Unix-like is required. Prior knowledge to numerical methods is recommended. The instructor uses English in the lectures and discussions.

Course design

How to study

Course schedule

Unit 1 Introduction

  1. Part 1: What is numerical software
    1. Why develop numerical software
    2. Hybrid architecture
    3. Numerical software = C++ + Python
  2. Part 2: What to learn in this course
    1. Term project
    2. How to write a proposal
    3. Term project grading guideline
    4. Online discussion
  3. Part 3: Runtime and course marterials
    1. Runtime environment: Linux and AWS
    2. Jupyter notebook

Unit 2 Fundamental engineering practices

A large chunk of efforts is spent in the infrastructure for coding. The key to the engineering system is automation.

  1. Automation
    1. Bash scripting
    2. Makefile
    3. Cmake (cross-platform, multi-language automation)
  2. Version control and regression
    1. Git version control system
    2. Automatic testing: author and run with google-test and py.test
    3. Wrap to Python and test there: pybind11
    4. Continuous integration to avoid regression
  3. Work that cannot be automated
    1. Code review (use github for demonstration)
    2. Timing to debug for performance
      1. Wall time and CPU cycles
      2. System time and user time
      3. Python timing tools

Unit 3 Python and numpy

Python is a popular choice for the scripting engine that makes the numerical software work as a platform.

The platform works like a library providing data structures and helpers to solve problems. The users will use Python to build applications.

  1. Organize Python modules
    1. Scripts
    2. Modules
    3. Packages
  2. Use numpy for array-oriented code
    1. Data type
    2. Construction
    3. Multi-dimensional arrays
    4. Selection
    5. Broadcasting
  3. Use tools for numerical analysis
    1. Matplotlib
    2. Linear algebra using numpy and scipy
    3. Package management wtih conda and pip

Unit 4 C++ and computer architecture

The low-level code of numerical software must be high-performance. The industries chose C++ because it can take advantage of everything that a hardware architecture offers while using any level of abstraction.

  1. Fundamental data types
    1. Command-line interface for compiler tools
      1. Compiler, linker
      2. Multiple source files, separation of declaration and definition, external libraries
      3. Build multiple binaries and shared objects (dynamically linked libraries)
    2. Integer, signness, pointer, array indexing
    3. Floating-point, rounding, exception handling
    4. Numeric limit
  2. Object-oriented programming
    1. Class, encapsulation, accessor, reference type
    2. Constructor and destructor
    3. Polymorphism and RTTI
    4. CRTP
  3. Standard template library (STL)
    1. std::vector, its internal and why the buffer address is dangerous
    2. std::array, std::list
    3. std::map, std::set, std::unordered_map, std::unordered_set

Unit 5 Matrix operations

Matrices are everywhere in numerical analysis. Arrays are the fundamental data structure and used for matrix-vector, matrix-matrix, and other linear algebraic operations.

  1. POD arrays and majoring
    1. Vector: 1D array
    2. Matrix: 2D array
    3. Row- and column-majoring
    4. A simple class for matrix
  2. Matrix-vector and matrix-matrix operations
    1. Matrix-vector multiplication
    2. Matrix-matrix multiplication
  3. Linear algebra
    1. Linear system solution
    2. Eigenvalue and singular value problems
    3. Least square problems

Unit 6 Cache optimization

How cache works, its importance to performance, and optimization with cache.

  1. Memory hierarchy
  2. How cache works
    1. Cache block (line) size determines speed
    2. Locality
    3. Matrix population in C++
    4. Array majoring in numpy
  3. Tiling

Unit 7 SIMD

Parallelism and x86 assembly for SIMD.

  1. Types of parallelism
    1. Shared-memory parallelism
    2. Distributed-memory parallelism
    3. Vector processing
  2. SIMD instructions
    1. CPU capabilities
    2. x86 intrinsic functions
    3. Symbol table
    4. Inspect assembly: 1, 3, 5 multiplications

Unit 8 Memory management

Numerical software tends to use as much memory as a workstation has. The memory has two major uses: (i) to hold the required huge amount of data, and (ii) to gain speed.

  1. Linux memory model: stack, heap, and memory map
  2. C memory management API
  3. C++ memory management API
  4. STL allocator API
  5. Object counter

Unit 9 Ownership and smart pointers

Ownership and memory management using C++ smart pointers.

  1. Pointers and ownership
    1. Raw pointer
    2. Reference
    3. Ownership
    4. Smart pointers
      1. unique_ptr
      2. shared_ptr
  2. Revisit shared pointer
    1. Make Data exclusively managed by shared_ptr
    2. Get shared_ptr from this
    3. Cyclic reference and weak_ptr

Unit 10 Modern C++

Copy elision and move semantics. Variadic template and perfect forwarding. Closure.

  1. Copy elision / return value optimization
    1. Make a tracker class for copy construction
    2. Show the copy elision in action
    3. Inspect the assembly
  2. Move semantics and copy elision
    1. Forced move is a bad idea
  3. Data concatenation
    1. Style 1: return vector
    2. Style 2: use output vector
    3. Style 3: use a class for both return and output argument
  4. Variadic template
  5. Perfect forwarding
  6. Lambda expression
    1. Keep a lambda in a local variable
    2. Difference between auto and std::function
  7. Closure
    1. Comments on functional style

Unit 11 C++ and C for Python

Use C++ and C to control the CPython interpreter.

  1. Expectation from Python
    1. Linear wave
    2. The inviscid Burgers equation
  2. Pybind11 build system
    1. Setuptools
    2. Cmake with a sub-directory
    3. Cmake with install pybind11
  3. Additional wrapping layer for customization
  4. Wrapping API
    1. Functions and property
    2. Named ane keyword arguments
    3. What happens in Python stays in Python (or pybind11)
  5. Manipulate Python objects in C++
  6. Python containers
    1. tuple
    2. list
    3. dict
  7. Use cpython API with pybind11
    1. PyObject reference counting
  8. Built-in types
    1. Cached value
    2. Attribute access
    3. Function call
    4. Tuple
    5. Dictionary
    6. List
  9. Useful operations
    1. Import
    2. Exception
  10. Python memory management
    1. PyMem interface
    2. Small memory optimization
    3. Tracemalloc

Unit 12 Array code in C++

Dissect the array-based code and element-based code and when to use them. Software architecture that take advantage of array-based code.

  1. Python is slow but easy to write
  2. Speed up by using numpy (still in Python)
  3. Xtensor: write iterative code in C++ speed using arrays
  4. Effect of house-keeping code
  5. Design interface with arrays
  6. Conversion between dynamic and static semantics
  7. Insert profiling code

Unit 12 Array-oriented design

Software architecture that take advantage of array-based code.

  1. Design interface with arrays
  2. Conversion between dynamic and static semantics
  3. Insert profiling code

Unit 13 Advanced Python

Advanced topics in Python programming.

  1. Iterator
    1. List comprehension
    2. Generator
    3. Generator expression
  2. Stack frame
    1. frame object
  3. Customizing module import with sys.meta_path
  4. Descriptor
    1. Keep data on the instance
  5. Metaclass
  6. Type introspection and abstract base class (abc)
    1. Method resolution order (mro)
    2. Abstract base class (abc)
    3. Abstract method


Command to bootstrap a server on AWS:

curl -s | bash -s setup_aws