Introduction to Mathematica Scientific Computing and Visualization Boston

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 To start Mathematica Kernel ( text-base interface), type math at the prompt: % math

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

Getting Started Notebook and Text-Based Interfaces Notebook Interface Text-Based Interface Start mathematica math Execute command Shift-Enter Exit Choose the Quit menu item Cntr-D or Quit[]

Numerical Calculations 21. 7 + 19. 94 Press Shift + Enter In[1] : = 21. 7 + 19. 94 Out[1] : = 41. 64

Numerical Calculations x+y+z add x-y subtract x/y divide x y z or x*y*z multiply x^y power x*(y+z) control grouping by parentheses Note: You can use space or a * sign for multiplication

Numerical Calculations You get exact result with Mathematica unless you request otherwise.

Numerical Calculations If an input number contains an explicit decimal point, Mathematica produces an approximate numerical result.

Numerical Calculations Common Mathematical Functions Sqrt[x] Exp[x] ex Log[x] ln x Log[b, x] logb x Sin[x], Cos[x], Tan[x] trigonometric functions Arc. Sin[x], . . . inverse trigonometric functions n! factorial Factor. Integer[n] prime factors of n Abs[x] |x| Round[x] closest integer to x Max[x, y, . . . ], Min[x, y, . . . ] maximum and minimum of a set Mod[n, m] remainder of division of n by m Random[] random number between 0 and 1

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

Numerical Calculations Common Mathematical Constants Pi E e Degree I Infinity ∞ The names of all built-in constants begin with capital letters.

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

Numerical Calculations Using Previous Results - use with care! • %% the next-to-last result • %n the result on output line Out[n] the last result generated 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!

Numerical Calculations Variables definition x = value x = y = value x =. or Clear[x] assign a value to the variable x assign a value to both x and y 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; • xy with no space means variable name xy; • 5 x means 5 times x;

Numerical Calculations Lists of Objects List is a collection of several objects in Mathematica

Graphics Partial list of Mathematica’s graphs • Graphics, Graphics 3 D • Plot, Plot 3 D • List. Plot, List. Line. Plot, List. Contour. Plot, List. Plot 3 D • List. Log. Plot, List. Polar. Plot, List. Surface. Plot 3 D, List. Contour. Plot 3 D • Prarametric. Plot, Polar. Plot, Revolution. Plot 3 D, Spherical. Plot 3 D, • Density. Plot, Relief. Plot • Graph. Plot, Array. Plot • Region. Plot, Contour. Plot, Region. Plot 3 D

Graphics Partial List of Graphics Options option name default value Aspect. Ratio 1/Golden. Ratio the height-to-width ratio for the plot; Axes True whether to include axes Axes. Label None labels to be put on the axes Frame False draw a frame around the plot Grid. Lines None what grid lines to include Plot. Label None an expression to be printed as a label for the plot Plot. Range Automatic the range of coordinates to include in the plot Ticks Automatic what tick marks to draw if there axes

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 • Chart. Element. Function[] for custom appearances

Graphics Feature Scope Options: • Specified globally and uniformly • Examples: Chart. Style, CHart. Labels, Labeling. Function, Background Wrappers: • Wrapped directly around data • Can be used at any level, allows for targeted use • Can be nested • Examples: Tooltip, Style, Button, Labeled

Data Visualization Basic Statistics Plots Bar Chart, Pie. Chart, Bubble. Chart Histogram, Smooth. Histogram, Density Histogram, Histogram 3 D Quantile. Plot, Probability. Scale. Plot Box. Whisker. Chart, Distribution. Chart

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.

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]

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}]

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 -> Arc. Sin[a]}}

Equations Solving equations You can also find an approximate numerical solution using Find. Root[]: In[1] : = Find. Root[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.

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

Code Optimization How to speedup Mathematica code Use Timing and Absolute. Timing commands to measure the time of execution: In[1] : = Module[{x = 1/Pi}, Do[x = 3. 5 x (1 - x), {10^6}]; x] //Absolute. Timing

Code Optimization How to speedup Mathematica code 1. Use floating point approximation if you can and as early in the code as possible 2. Compile Functions 3. Use parallelization options if possible 4. Use built-in functions 5. Use Cuda. Link if possible

HPC: Vectorization Automatic parallelization Making a Compiled Function run in parallel is simple - the user only has to pass the option Runtime. Attributes->"Listable". From there, Compile will run in as many threads as there are on the system. ray. Spheres. Intersection. Color = Compile[. . . , Compilation. Target -> "C", Runtime. Attributes -> Listable]; ray. Trace. Compile[. . . , ray. Spheres. Intersection. Color[] ]; width = height = 300; . . . imgc = ray. Trace. Compile[centers, radii, colors, x, y]; Image[imgc]

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

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

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 : 1. NVIDIA hardware 2. Recent video card driver 3. Supported C – compiler

HPC: GPU Programming 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 M 2070", "Clock Rate" -> 1147000, "Compute Capabilities" -> 2. , "GPU Overlap" -> 1, "Maximum Block Dimensions" -> {1024, 64}, "Maximum Grid Dimensions" -> {65535, 65535},

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

HPC: GPU Programming CUDALink Image Processing In[1] : = swan = Import["swan. JPG"] Out[1] : = In[2] : = CUDAImage. Convolve[swan, Gaussian. Matrix[16]] (* Convolution *) Out[2] : =

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