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Math, Numerics, and Programming (for Mechanical Engineers)

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Math, Numerics, and Programming (for Mechanical Engineers)

This text introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches.

Mechanical EngineeringEngineering
Author
Masayuki Yano, James Douglass Penn, George Konidaris, & Anthony T Patera
Publisher
Massachusetts Institute of Technology
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Information Technology
Read online Download PDF (open licence)
Contents· 118 sections
  1. 01: Unit I - (Numerical) Calculus. Elementary Programming Concepts
  2. 01: Motivation
  3. 1.01: A Mobile Robot
  4. 1.02: Global Position Estimation- Infra-red Range-Finding
  5. 1.03: Local Position Tracking - Odometry
  6. 1.04: The Numerical Tasks
  7. 02: Interpolation
  8. 2.01: Interpolation of Univariate Functions
  9. 2.02: Interpolation of Bivariate Functions
  10. 03: Differentiation
  11. 3.01: Differentiation of Univariate Functions
  12. 3.02: Differentiation of Bivariate Functions
  13. 04: Elements of a Program and Matlab Basics
  14. 4.01: Computer Architecture and Computer Programming
  15. 4.02: Data Types (and Classes)
  16. 4.03: Variables and Assignment
  17. 4.04: The Workspace and Saving Loading Data
  18. 4.05: Arithmetic Operations
  19. 4.06: Floating Point Numbers (FPNs): Representation and Operations
  20. 4.07: Relational and Logical Operations
  21. 4.08: Flow Control
  22. 05: Matlab Arrays
  23. 5.01: Single-Index Floating Point Arrays
  24. 5.02: Characters and Character Single-Index Arrays (Strings)
  25. 5.03: Double-Index Arrays
  26. 5.04: Line Plotting
  27. 06: Functions in Matlab
  28. 6.01: The Advantage - Encapsulation and Re-Use
  29. 6.02: Always Test a Function
  30. 6.03: What Happens in a Function Stays in a Function
  31. 6.04: Syntax - Inputs (Parameters) and Outputs
  32. 6.05: Functions of Functions - Handles
  33. 6.06: Anonymous (or In-Line) Functions
  34. 6.07: String Inputs and the eval Function
  35. 07: Integration
  36. 7.01: Integration of Univariate Functions
  37. 7.02: Integration of Bivariate Functions
  38. 02: Unit II - Monte Carlo Methods
  39. 08: Introduction
  40. 8.01: Statistical Estimation and Simulation
  41. 8.02: Motivation - An Example
  42. 09: Introduction to Random Variables
  43. 9.01: Discrete Random Variables
  44. 9.02: Discrete Bivariate Random Variables (Random Vectors)
  45. 9.03: Binomial Distribution
  46. 9.04: Continuous Random Variables
  47. 9.5: Continuous Random Vectors
  48. 10: Statistical Estimation- the Normal Density
  49. 10.01: Introduction
  50. 10.02: The Sample Mean - An Estimator Estimate
  51. 10.03: Confidence Intervals
  52. 10.04: Cumulative Sample Means
  53. 11: Monte Carlo- Areas and Volumes
  54. 12: Monte Carlo- General Integration Procedures
  55. 12.01: Calculating an Area
  56. 12.02: Calculation of Volumes in Higher Dimensions
  57. 13: Statistical Estimation- the Normal Density
  58. 14: Monte Carlo- Failure Probabilities
  59. 14.01: Calculating a Failure Probability
  60. 03: Unit III - Linear Algebra 1 - Matrices Least Squares and Regression
  61. 15: Motivation
  62. 16: Matrices and Vectors - Definitions and Operations
  63. 16.01: Basic Vector and Matrix Operations
  64. 16.02: Matrix Operations
  65. 16.03: Special Matrices
  66. 16.04: Further Concepts in Linear Algebra
  67. 17: Least Squares
  68. 17.01: Data Fitting in Absence of Noise and Bias
  69. 17.02: Overdetermined Systems
  70. 17.03: Least Squares
  71. 18: Matlab Linear Algebra (Briefly)
  72. 18.01: Matrix Multiplication (and Addition)
  73. 18.02: The MATLAB Inverse Function - inv
  74. 18.03: Solution of Linear Systems - MATLAB Backslash
  75. 18.04: Solution of (Linear) Least-Squares Problems
  76. 19: Regression - Statistical Inference
  77. 19.01: Simplest Case
  78. 19.02: General Case
  79. 04: Unit IV - (Numerical) Differential Equations
  80. 20: Motivation
  81. 21: Initial Value Problems
  82. 21.01: Scalar First-Order Linear ODEs
  83. 21.02: Scalar Second-Order Linear ODEs
  84. 21.03: System of Two First-Order Linear ODEs
  85. 21.04: IVPs - System of n Linear ODEs
  86. 22: Boundary Value Problems
  87. 23: Partial Differential Equations
  88. 05: Unit V - (Numerical) Linear Algebra 2 - Solution of Linear Systems
  89. 24: Motivation
  90. 24.1: A Robot Arm
  91. 24.2: Gaussian Elimination and Sparsity
  92. New Page
  93. 25: Linear Systems
  94. 25.01: Model Problem- n 2 Spring-Mass System in Equilibrium
  95. 25.02: Existence and Uniqueness- n 2
  96. 25.03: New Page
  97. 25.04: Existence and Uniqueness - General Case (Square Systems)
  98. 26: Gaussian Elimination and Back Substitution
  99. 26.01: A 2 2 System (n 2)
  100. 26.02: A 3 3 System (n 3)
  101. 26.3: General (n times n) Systems
  102. 26.4: Gaussian Elimination and LU Factorization
  103. New Page
  104. 27: Gaussian Elimination - Sparse Matrices
  105. 27.01: Banded Matrices
  106. 27.02: Matrix-Vector Multiplications
  107. 27.03: Gaussian Elimination and Back Substitution
  108. 27.04: Fill-in and Reordering
  109. 27.05: The Evil Inverse
  110. 28: Sparse Matrices in Matlab
  111. 28.01: The Matrix Vector Product
  112. 28.02: Sparse Gaussian Elimination
  113. 06: Unit VI - Nonlinear Equations
  114. 29: Newton Iteration
  115. 29.01: Introduction
  116. 29.02: Univariate Newton
  117. 29.03: Multivariate Newton
  118. 29.04: Continuation and Homotopy