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Introduction to Mathematica

Introduction to Mathematica. Scientific Computing and Visualization Boston University Katia Oleinik koleinik@bu.edu. Getting Started. Notebook and Text-Based Interfaces To start Mathematica on SCC cluster, type mathematica (or Mathematica ) at the prompt: % mathematica

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Introduction to Mathematica

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  1. Introduction to Mathematica Scientific Computing and Visualization Boston University Katia Oleinikkoleinik@bu.edu

  2. Getting Started Notebook and Text-Based Interfaces To start Mathematica on SCC cluster, type mathematica(or Mathematica) at the prompt: % mathematica To start Mathematica Kernel ( text-base interface), type math at the prompt: % math

  3. Getting Started Notebook and Text-Based Interfaces To start Mathematica on Windows: Start->Wolfram Mathematica ->Wolfram Mathematica 9 To start Mathematica Kernel ( text-base interface) on Windows: Start->Wolfram Mathematica ->Wolfram Mathematica 9 Kernel

  4. Getting Started Notebook and Text-Based Interfaces

  5. Numerical Calculations 21.7 + 19.94 Press Shift + Enter In[1] := 21.7 + 19.94 Out[1] := 41.64

  6. Numerical Calculations Note: You can use space or a * sign for multiplication

  7. Numerical Calculations You get exactresult with Mathematica unless you request otherwise. In[1] := 2 ^ 100 (* get exact result *) Out[1] := 1267650600228229401496703205376 In[2] := 2 ^ 100 //N (* get approximation *) Out[2] := 1.26765x1030 In[3] := 1/3 + 2/7 (* get exact result *) Out[3] := In[4] := 1/3 + 2/7 //N (* get approximation *) Out[4] := 0.619048

  8. Numerical Calculations If an input number contains an explicit decimal point, Mathematica produces an approximate numerical result. In[5] := 11/3 + 2/7 (* exact result *) Out[5] := In[6] := 1.1/3 + 2/7 (* approximation *) Out[6] := 0.652381

  9. Numerical Calculations Common Mathematical Functions

  10. Numerical Calculations Functions in Mathematica • The arguments of ALL Mathematica functions are enclosed in square brackets; • The names of built-in Mathematica functions begin with capital letters; • Unless //N option or decimal point is present, Mathematica tries to output exact value

  11. Numerical Calculations Common Mathematical Constants The names of all built-in constants begin with capital letters.

  12. Numerical Calculations Specify the degree of precision In[1] := N[Pi, 30] (* approximation *) Out[1] := 3.14159265358979323846264338328 In[2] := N[Sqrt[7], 10] Out[2] := 2.645751311

  13. Numerical Calculations Using Previous Results - use with care! • % the last result generated • %% the next-to-last result • %n the result on output line Out[n] In[1] := 7 + 3 Out[1] := 10 In[2] := % + 1 Out[2] := 11 Note: % is always defined to be the last result that Mathematica generated. It can be anywhere in the script!

  14. Numerical Calculations Variables definition x = valueassign a value to the variable x x = y = valueassign a value to both x and y x = . orClear[x]remove any value assigned to x • Notes: • Mathematica is case-sensitive; • To avoid confusion with built-in functions, choose names that start with lower-case letters; • x y means x times y; • xywith no space means variable name xy; • 5x means 5 times x;

  15. Numerical Calculations Lists of Objects List is a collection of several objects in Mathematica In[1] := vec = {2, 4, 1.8} Out[1] := {2, 4, 1.8} In[2] := vec^2 Out[2] := {4, 16, 3.24} In[3] := vec/(vec-1) Out[3] := {1, , 2.25} In[4] := vec[[2]] (* extract second element *) Out[4] := 4 In[5] := Part[vec,1] (* extract first element *) Out[5] := 2

  16. Graphics Partial list of Mathematica’s graphs • Graphics, Graphics3D • Plot, Plot3D • ListPlot, ListLinePlot, ListContourPlot, ListPlot3D • ListLogPlot, ListPolarPlot, ListSurfacePlot3D, ListContourPlot3D • PrarametricPlot, PolarPlot, RevolutionPlot3D, SphericalPlot3D, • DensityPlot, ReliefPlot • GraphPlot, ArrayPlot • RegionPlot, ContourPlot, RegionPlot3D

  17. Graphics Partial List of Graphics Options

  18. Graphics Basic Features for Visualization Functions • Feature Classes: • Styles • Colors, thickness, pointsize, opacity, … • Element appearance and shape • Labels • Textual labels and tooltips • Legends • Interactions • Built-in highlighting effects • Use elements for buttons, popup windows and other events • Metadata • Wrapper used to include additional information • ChartElementFunction[] for custom appearances

  19. Graphics Feature Scope • Options: • Specified globally and uniformly • Examples: ChartStyle, CHartLabels, LabelingFunction, Background • Wrappers: • Wrapped directly around data • Can be used at any level, allows for targeted use • Can be nested • Examples: Tooltip, Style, Button, Labeled

  20. Data Visualization Basic Statistics Plots • Bar Chart, PieChart, BubbleChart • Histogram, SmoothHistogram, Density Histogram, Histogram3D • QuantilePlot, • ProbabilityPlot, ProbabilityScalePlot • BoxWhiskerChart, • DistributionChart

  21. Functions Function Definition • Use underscore _ after a variable name (function argument) • f([x_] := x^2 + 4 x + 4 • To execute a function, simply call it with a given value: • f[1] • Mathematica’s built-in functions start with upper-case letter. Start with lower case letter for the user defined function.

  22. Functions Using Functions Show function definition: ?f Expend function: Expand[f[x + y + 1]] Find derivative of a function: D[f[x], x] Find integral of a function: Integrate[f[x] ,x] Find definite integral of a function: Integrate[f[x] ,{x,0,1}] Clear the definition of a function: Clear[f]

  23. Equations Solving equations • Define equation using double equal sign == • Solve[4 x^2 + 4 x + 1 == 0] • Mathematica can solve an equation for one variable in terms of another • Solve[5 x^2 -2 Log[y] == 3 x,y] • Solve system of equations: • Solve[{x + 2 y == 5, 7 x – 5 x == -3},{x,y}]

  24. Equations Solving equations Mathematica can solve algebraic equations in one variable for power less than 5 and sometimes even higher. But there are some equations for which it is impossible to find the root(s) algebraically. Mathematica will use Root object to represent the solution. Use N[%] to evaluate the solution numerically. In some cases Mathematica can solve equations involving other functions: In[1] := Solve[Sin[x] == a, x] Out[1] := {{ x -> ArcSin[a]}}

  25. Equations Solving equations You can also find an approximate numerical solution using FindRoot[]: In[1] := FindRoot[Cos[x] == x, {x,0}] Out[1] := { x -> 0.739085} Mathematica can solve system of simultaneous equations. It can eliminate a variable in a system, using Eliminate[] function, or simplify the system using Reduce[] function.

  26. Programs Programming Constructs Assignments: = += ++ *= AppendTo Loops: Do While For Table Nest Conditionals: If Which Switch And(&&) Equal(==) Less(<) … Flow Control: Return Throw Catch TimeConstrained Scope Constructs: Module With Block I/O: Print Input Pause Import OpenRead …

  27. Code Optimization How to speedup Mathematica code Use Timing and AbsoluteTiming commands to measure the time of execution: In[1] := Module[{x = 1/Pi}, Do[x = 3.5 x (1 - x), {10^6}]; x] //AbsoluteTiming

  28. Code Optimization How to speedup Mathematica code Use floating point approximation if you can and as early in the code as possible Compile Functions Use parallelization options if possible Use built-in functions Use CudaLink if possible

  29. HPC: Vectorization Automatic parallelization Making a Compiled Function run in parallel is simple - the user only has to pass the option RuntimeAttributes->"Listable". From there, Compile will run in as many threads as there are on the system. raySpheresIntersectionColor = Compile[ . . . , CompilationTarget -> "C", RuntimeAttributes-> Listable]; rayTraceCompile[ . . . , raySpheresIntersectionColor[] ]; width = height = 300; . . . imgc = rayTraceCompile[centers, radii, colors, x, y]; Image[imgc]

  30. HPC: GPU Programming CUDA and OpenCL • Programming languages/environments that allow to write to program GPU to perform general computations. • CUDA works only on NVIDIA hardware and is proprietary • OpenCL works on AMD and NVIDIA hardware and is an open standard

  31. HPC: GPU Programming Why GPU Speed! Modern GPUs are capable to perform 3 TFlops/sec, while high end CPUs – 80GFlops/sec Fastest supercomputer at the end on 90s, clocked 1TFlop/sec and now you can buy GPU with the same speed for less than $500. Speed comes from the hardware design.

  32. HPC: GPU Programming CUDALink • A way to use the Graphical Processing Unit (GPU) from within Mathematica integrating with the user’s workflow • A way to load GPU programs into Mathematica and use them as functions • It is NOT an attempt to make all Mathematica functions utilize the GPU • It is NOT meant to automatically speed up Mathematica code • Require : • NVIDIA hardware • Recent video card driver • Supported C – compiler

  33. HPC: GPU Programming CUDALink CUDALink contains many built-in function for performing linear algebra, list and image processing, and Fourier analysis. In[1] := Needs["CUDALink`"] (* Load CUDALink *) In[2] := CUDAQ[] (* Check if system compatible with CUDA *) Out[2] := TRUE In[3] := CUDAInformation[](* Get information about detected GPU *) Out[3] := {1 -> {"Name" -> "Tesla M2070", "Clock Rate" -> 1147000, "Compute Capabilities" -> 2., "GPU Overlap" -> 1, "Maximum Block Dimensions" -> {1024, 1024, 64}, "Maximum Grid Dimensions" -> {65535, 65535, 65535},

  34. HPC: GPU Programming CUDALink OnceCUDALinkis loaded built-in functions can be used: In[1] := m1 = {{1, 1, 1}, {2, 2, 2}, {3, 3, 3}} In[2] := m2 = {{0, 1, 0}, {1, 1, 1}, {0, 1, 0}} In[3] := CUDADot[m1,m2] (* Multiply matricies*) Out[2] := {{1, 3, 1}, {2, 6, 2}, {3, 9, 3}}

  35. HPC: GPU Programming CUDALink Image Processing In[1] := swan = Import["swan.JPG"] Out[1] := In[2] := CUDAImageConvolve[swan, GaussianMatrix[16]] (* Convolution *) Out[2] :=

  36. This tutorial has been made possible by Scientific Computing and Visualization group at Boston University. Katia Oleinikkoleinik@bu.edu http://www.bu.edu/tech/research/training/tutorials/list/

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