# Introduction to Mathematica Scientific Computing and Visualization Boston University Katia Oleinik

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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 -> Wolfram Mathematica 9 To start Mathematica Kernel ( text-base interface) on Windows: Start -> Wolfram Mathematica -> Wolfram Mathematica 9 Kernel

Getting Started Notebook and Text-Based Interfaces Notebook InterfaceText-Based Interface Startmathematicamath Execute commandShift-EnterEnter ExitChoose the Quit menu itemCntr-D or Quit[]

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

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 Sqrt[x] Exp[x] e x Log[x] ln x Log[b,x] log b x Sin[x], Cos[x], Tan[x] trigonometric functions ArcSin[x],... inverse trigonometric functions n! factorial FactorInteger[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 Common Mathematical Functions

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 Pi E e Degree I Infinity ∞ Common Mathematical Constants 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! In[1] := 7 + 3 Out[1] := 10 In[2] := % + 1 Out[2] := 11 % the last result generated % the next-to-last result %n the result on output line Out[n] 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 assign a value to the variable x x = y = value assign a value to both x and y x =. or Clear[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; xy with no space means variable name xy; 5x 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, Graphics3D Plot, Plot3D ListPlot, ListLinePlot, ListContourPlot, ListPlot3D ListLogPlot, ListPolarPlot, ListSurfacePlot3D, ListContourPlot3D PrarametricPlot, PolarPlot, RevolutionPlot3D, SphericalPlot3D, DensityPlot, ReliefPlot GraphPlot, ArrayPlot RegionPlot, ContourPlot, RegionPlot3D

Graphics Partial List of Graphics Options option namedefault value AspectRatio1/GoldenRatio the height-to-width ratio for the plot; AxesTrue whether to include axes AxesLabelNone labels to be put on the axes FrameFalse draw a frame around the plot GridLinesNone what grid lines to include PlotLabelNone an expression to be printed as a label for the plot PlotRangeAutomatic the range of coordinates to include in the plot TicksAutomatic what tick marks to draw if there are 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 ChartElementFunction[] for custom appearances

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

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

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 -> ArcSin[a]}}

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.

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 …

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

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 CudaLink if possible

HPC: Vectorization 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. Automatic parallelization raySpheresIntersectionColor = Compile[..., CompilationTarget -> "C", RuntimeAttributes -> Listable]; rayTraceCompile[..., raySpheresIntersectionColor[] ]; width = height = 300;... imgc = rayTraceCompile[centers, radii, colors, x, y]; Image[imgc]

HPC: GPU Programming 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 CUDA and OpenCL

HPC: GPU Programming 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. Why GPU

HPC: GPU Programming 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 CUDALink

HPC: GPU Programming CUDALink contains many built-in function for performing linear algebra, list and image processing, and Fourier analysis. CUDALink 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},

HPC: GPU Programming Once CUDALink is loaded built-in functions can be used: CUDALink 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}}

HPC: GPU Programming Image Processing CUDALink In[1] := swan = Import["swan.JPG"] Out[1] := In[2] := CUDAImageConvolve[swan, GaussianMatrix[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/

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