Overview: The original Mathematica was a computer algebra system (CAS) released by Stephen Wolfram in 1988.

Transcription

1 Mathematica Overview: The original Mathematica was a computer algebra system (CAS) released by Stephen Wolfram in Modern releases have augmented the CAS with powerful numerical and graphical capabilities. Emphasis is placed on allowing math to be incorporated into neat final products - through fancy formatting, interactivity etc. Mathematica is renowned as the world s ultimate application for computations. But it s much more - it s the only development platform fully integrating computation into complete workflows, moving you seamlessly from initial ideas all the way to deployed individual or enterprise solutions.

2 Mathematica It s clear who Wolfram is ultimately trying to entice: 100% of the Fortune 50 companies rely on Mathematica to maintain their competitive edge in innovation.

3 Mathematica It s clear who Wolfram is ultimately trying to entice: 100% of the Fortune 50 companies rely on Mathematica to maintain their competitive edge in innovation. I mainly use the basic CAS only, for: simplification differentiation integration (quick) visualization This talk focuses on the background needed to execute these tasks efficiently.

4 Mathematica GUI: Mathematica (.nb) files are called notebooks. A notebook consists of input and output cells.

5 Mathematica GUI: Mathematica (.nb) files are called notebooks. A notebook consists of input and output cells. Each input cell contains at least one command. Evaluating an input cell (Shift-Enter) produces an output cell immediately below it.

6 Mathematica GUI: Mathematica (.nb) files are called notebooks. A notebook consists of input and output cells. Each input cell contains at least one command. Evaluating an input cell (Shift-Enter) produces an output cell immediately below it. Matlab analogy: Notebooks mix elements from m-files (saving/loading of multiple commands) and the interactive command window (evaluation cell-by-cell).

7 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= /3 2*3 2^3

8 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= /3 2*3 2^3 Evaluate whole cell by pressing Shift-Enter

9 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= /3 2*3 2^3 Out[1]= 5 Out[2]= -1 Out[3]= 2/3 Out[4]= 6 Out[5]= 8

10 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= /3 2*3 2^3 Out[1]= 5 Out[2]= -1 Out[3]= 2/3 Out[4]= 6 Out[5]= 8 We can modify our input to return a decimal using the N[...] function.

11 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= N[2/3] 2*3 2^3 Alter this line only... Out[1]= 5 Out[2]= -1 Out[3]= 2/3 Out[4]= 6 Out[5]= 8

12 Mathematica 7.0.1: Basics Basic mathematical operations: In[1]:= N[2/3] 2*3 2^3...then re-evaluate this whole cell with Shift-Enter Out[1]= 5 Out[2]= -1 Out[3]= 2/3 Out[4]= 6 Out[5]= 8

13 Mathematica 7.0.1: Basics Basic mathematical operations: In[6]:= N[2/3] 2*3 2^3 Out[6]= 5 Out[7]= -1 Out[8]= Out[9]= 6 Out[10]= 8 The new input and output cells visually replace the old ones.

14 Mathematica 7.0.1: Basics Storing/accessing output: Variable assignments are straightforward: In[1]:= a=3 a+4 Out[1]= 3 Out[2]= 7

15 Mathematica 7.0.1: Basics Storing/accessing output: Variable assignments are straightforward: In[1]:= a=3 a+4 Out[1]= 3 Out[2]= 7 The most recent output value is stored in % (like Matlab s ans): In[3]:= %+5 Out[3]= 12

16 Mathematica 7.0.1: Basics Storing/accessing output: Variable assignments are straightforward: In[1]:= a=3 a+4 Out[1]= 3 Out[2]= 7 The most recent output value is stored in % (like Matlab s ans): In[3]:= %+5 Out[3]= 12 Earlier output values are accessed using Out[...]: In[4]:= %+Out[1] Out[4]= 15

17 Mathematica 7.0.1: Basics General syntax rules: Constants and built-in functions are always capitalized. Multi-word functions use multiple capital letters. Function arguments go inside brackets. Functions can be nested.

18 Mathematica 7.0.1: Basics General syntax rules: Constants and built-in functions are always capitalized. Multi-word functions use multiple capital letters. Function arguments go inside brackets. Functions can be nested. Examples: Constants: E, I, Pi, Infinity. Basic math functions: N[...], Exp[...], Log[...], Sin[...], Cos[...], Tan[...], Abs[...], etc. Manipulation functions: Factor[...], Expand[...], FullSimplify[...], TeXForm[...].

19 Mathematica 7.0.1: Functions Built-in function use: In[1]:= N[Cos[2]] Out[1]=

20 Mathematica 7.0.1: Functions Built-in function use: In[1]:= N[Cos[2]] Out[1]= In[2]:= Expand[(-10 + x)(7 + x)] Out[2]= -70-3x + x^2 Note: * operator only needed to avoid ambiguity!

21 Mathematica 7.0.1: Functions Built-in function use: In[1]:= N[Cos[2]] Out[1]= In[2]:= Expand[(-10 + x)(7 + x)] Out[2]= -70-3x + x^2 Note: * operator only needed to avoid ambiguity! In[3]:= TeXForm[x/(x+1)] Out[3]= \frac{x}{x+1}

22 Mathematica 7.0.1: Functions Misc. syntax tips: To get help on a function use?functionname or??functionname: In[1]:=?Cos Out[1]= Cos[z] gives the cosine of z.

23 Mathematica 7.0.1: Functions Misc. syntax tips: To get help on a function use?functionname or??functionname: In[1]:=?Cos Out[1]= Cos[z] gives the cosine of z. Incorrectly-typed function names appear blue: In[2]:= texform[x/(x+1)] TeXForm[x/(x+1)] Out[2]= texform[x/(x+1)] Out[3]= \frac{x}{x+1}

24 Mathematica 7.0.1: Functions Misc. syntax tips: To get help on a function use?functionname or??functionname: In[1]:=?Cos Out[1]= Cos[z] gives the cosine of z. Incorrectly-typed function names appear blue: In[2]:= texform[x/(x+1)] TeXForm[x/(x+1)] Out[2]= texform[x/(x+1)] Out[3]= \frac{x}{x+1} As in Matlab, output is suppressed by a semi-colon: In[4]:= 2+3; Out[4] is defined, just hidden.

25 Mathematica 7.0.1: Functions User-defined functions: In[1]:= f[x_]:=x^2 Delayed assignment - no output.

26 Mathematica 7.0.1: Functions User-defined functions: In[1]:= f[x_]:=x^2 Delayed assignment - no output. In[2]:= f[3] f[t] Out[2]= 9 Out[3]= t^2

27 Mathematica 7.0.1: Functions User-defined functions: In[1]:= f[x_]:=x^2 Delayed assignment - no output. In[2]:= f[3] f[t] Out[2]= 9 Out[3]= t^2 In[4]:= g[x_,y_]:=abs[x-y] g[-3,3] g[s+h,s-h] Out[5]= 6 Out[6]= 2 Abs[h]

28 Mathematica 7.0.1: Functions (Advanced) Patterns are used to restrict domains: In[1]:= g[x_?integerq]:=x^2 g[2] g[2.5] Out[2]= 4 Out[3]= g[2.5]

29 Mathematica 7.0.1: Functions (Advanced) Patterns are used to restrict domains: In[1]:= g[x_?integerq]:=x^2 g[2] g[2.5] Out[2]= 4 Out[3]= g[2.5] In[4]:= h[x_?numericq]:=x^3 h[2] h[2.5] h[t] Out[5]= 8 Out[6]= Out[7]= h[t] Sometimes needed for plotting. Patterns will only redefine a function on the restricted domain.

30 Mathematica 7.0.1: Lists Lists: Syntax: {1,3,9,27}

31 Mathematica 7.0.1: Lists Lists: Syntax: {1,3,9,27} Elements are extracted using double brackets: In[1]:= {1,3,9,27}[[2]] Out[1]= 3

32 Mathematica 7.0.1: Lists Lists: Syntax: {1,3,9,27} Elements are extracted using double brackets: In[1]:= {1,3,9,27}[[2]] Out[1]= 3 Nesting forms arrays: In[2]:= {{1,3},{9,27}}[[2,1]] Out[2]= 9

33 Mathematica 7.0.1: Lists Lists: Syntax: {1,3,9,27} Elements are extracted using double brackets: In[1]:= {1,3,9,27}[[2]] Out[1]= 3 Nesting forms arrays: In[2]:= {{1,3},{9,27}}[[2,1]] Out[2]= 9 The Range[...] function can be used to generate lists: In[3]:= Range[1,9,2] Out[3]= {1,3,5,7,9}

34 Mathematica 7.0.1: Lists Lists as inputs: In[1]:= Sin[{0,Pi/2}] Out[1]= {0,1}

35 Mathematica 7.0.1: Lists Lists as inputs: In[1]:= Sin[{0,Pi/2}] Out[1]= {0,1} In[2]:= N[Cos[Range[1,3]]] Out[2]= { , , }

36 Mathematica 7.0.1: Lists Lists as inputs: In[1]:= Sin[{0,Pi/2}] Out[1]= {0,1} In[2]:= N[Cos[Range[1,3]]] Out[2]= { , , } Lists as options: In[3]:= Sum[n,{n,1,9,2}] Sum of odd #s between 1 and 9. Out[3]= 25

37 Mathematica 7.0.1: Lists Lists as inputs: In[1]:= Sin[{0,Pi/2}] Out[1]= {0,1} In[2]:= N[Cos[Range[1,3]]] Out[2]= { , , } Lists as options: In[3]:= Sum[n,{n,1,9,2}] Sum of odd #s between 1 and 9. Out[3]= 25 In[4]:= Series[Exp[w],{w,0,2}] Taylor series of Exp[w] about Out[4]= 1 + w + w^2/2 + O[w^3] w = 0, up to order 2.

38 Mathematica 7.0.1: Differentiation Differentiation: Of undefined functions: In[1]:= D[f[x],x] D[f[x],{x,2}] Out[1]= f [x] Out[2]= f [x]

39 Mathematica 7.0.1: Differentiation Differentiation: Of undefined functions: In[1]:= D[f[x],x] D[f[x],{x,2}] Out[1]= f [x] Out[2]= f [x] Of given functions: In[3]:= g[x_]:=exp[-x^2] D[g[t],t] D[g[t],{t,2}] Out[4]= -2 t Exp[-t^2] Out[5]= -2 Exp[-t^2] + 4 t^2 Exp[-t^2]

40 Mathematica 7.0.1: Differentiation To define the resulting expression as a new function, use immediate assignment: In[1]:= h[t_]=d[exp[-t^2],{t,2}] h[0] Out[1]= -2 Exp[t^2] + 4 t Exp[t^2] Out[2]= -2

41 Mathematica 7.0.1: Differentiation To define the resulting expression as a new function, use immediate assignment: In[1]:= h[t_]=d[exp[-t^2],{t,2}] h[0] Out[1]= -2 Exp[t^2] + 4 t Exp[t^2] Out[2]= -2 Delayed assignment will not work here! In[3]:= k[t_]:=d[exp[-t^2],{t,2}] k[0] General::ivar: 0 is not a valid variable. Out[4]= D[1,{0,2}]

42 Mathematica 7.0.1: Immediate/Delayed Assignment Short version: Use := to define functions with known expressions. Use = to define functions resulting from other manipulations, and all constants.

43 Mathematica 7.0.1: Immediate/Delayed Assignment Short version: Use := to define functions with known expressions. Use = to define functions resulting from other manipulations, and all constants. (Advanced) Longer version: Immediate assignment (f[x_]=...) evaluates the RHS expression once (when first called) and assigns the result to f[x] forever. Delayed assignment (f[x_]:=...) evaluates the RHS expression each time f is called. The value of x is substituted into the RHS expression before all algebraic and numerical manipulations are evaluated.

44 Mathematica 7.0.1: Integration Integration: In[1]:= Integrate[t^2,{t,1,2}] Integrate[Cos[t],{t,0,x}] Integrate[Exp[-t^2],{t,0,Infinity}] Integrate[Exp[-t^2],{t,0,1}] Out[1]= 7/3 Out[2]= Sin[x] Out[3]= Sqrt[Pi]/2 Out[4]= (Sqrt[Pi] Erf[1])/2

45 Mathematica 7.0.1: Integration Integration: In[1]:= Integrate[t^2,{t,1,2}] Integrate[Cos[t],{t,0,x}] Integrate[Exp[-t^2],{t,0,Infinity}] Integrate[Exp[-t^2],{t,0,1}] Out[1]= 7/3 Out[2]= Sin[x] Out[3]= Sqrt[Pi]/2 Out[4]= (Sqrt[Pi] Erf[1])/2 For integrals with no closed-form result, use NIntegrate: In[5]:= NIntegrate[Exp[-t^2],{t,0,1}] Out[5]= Quite a few functions have a numerical equivalent with similar syntax.

46 Mathematica 7.0.1: Integration Mathematica can (usually) handle ambiguous cases: In[1]:= Integrate[t^n,{t,1,Infinity}] Out[1]= ConditionalExpression[-1/(1+n), Re[n] < -1]

47 Mathematica 7.0.1: Integration Mathematica can (usually) handle ambiguous cases: In[1]:= Integrate[t^n,{t,1,Infinity}] Out[1]= ConditionalExpression[-1/(1+n), Re[n] < -1] It is also possible to build assumptions in: In[2]:= Integrate[t^n,{t,0,1},Assumptions->{Re[n] > -1}] Out[2]= 1/(1+n)

48 Mathematica 7.0.1: Integration Mathematica can (usually) handle ambiguous cases: In[1]:= Integrate[t^n,{t,1,Infinity}] Out[1]= ConditionalExpression[-1/(1+n), Re[n] < -1] It is also possible to build assumptions in: In[2]:= Integrate[t^n,{t,0,1},Assumptions->{Re[n] > -1}] Out[2]= 1/(1+n) Iterated integrals are performed from right to left: In[3]:= Integrate[1,{x,0,1},{y,0,x}] Out[3]= 1/2

49 Mathematica 7.0.1: Transformation Rules Transformation Rules: Assumptions->{Re[n] > -1} is called a transformation rule.

50 Mathematica 7.0.1: Transformation Rules Transformation Rules: Assumptions->{Re[n] > -1} is called a transformation rule. These are frequently found in function options or as output from equation-solving functions.

51 Mathematica 7.0.1: Transformation Rules Transformation Rules: Assumptions->{Re[n] > -1} is called a transformation rule. These are frequently found in function options or as output from equation-solving functions. Transformation rules always have the form Variable->Value, e.g. x->2.

52 Mathematica 7.0.1: Transformation Rules Transformation Rules: Assumptions->{Re[n] > -1} is called a transformation rule. These are frequently found in function options or as output from equation-solving functions. Transformation rules always have the form Variable->Value, e.g. x->2. Rules are applied to expressions using the /. operator: In[1]:= 3^x/.x->2 Out[1]= 9

53 Mathematica 7.0.1: Solving Equations Solving algebraic equations: Solve[eqns,vars] solves the list of polynomial equations/inequalities eqns for the list of variables vars: In[1]:= Solve[x^2+1 == 0,x] Out[1]= {{x -> -I}, {x -> I}}

54 Mathematica 7.0.1: Solving Equations Solving algebraic equations: Solve[eqns,vars] solves the list of polynomial equations/inequalities eqns for the list of variables vars: In[1]:= Solve[x^2+1 == 0,x] Out[1]= {{x -> -I}, {x -> I}} All input equalities are written using ==.

55 Mathematica 7.0.1: Solving Equations Solving algebraic equations: Solve[eqns,vars] solves the list of polynomial equations/inequalities eqns for the list of variables vars: In[1]:= Solve[x^2+1 == 0,x] Out[1]= {{x -> -I}, {x -> I}} All input equalities are written using ==. Solutions are returned as transformation rules: In[2]:= x/.out[1] Out[2]= {-I,I}

56 Mathematica 7.0.1: Solving Equations Solving algebraic equations: Solve[eqns,vars,dom] allows solution over restricted domains: In[1]:= Solve[x^2+1 == 0,x,Reals] Out[1]= {} (or Integers.)

57 Mathematica 7.0.1: Solving Equations Solving algebraic equations: Solve[eqns,vars,dom] allows solution over restricted domains: In[1]:= Solve[x^2+1 == 0,x,Reals] Out[1]= {} (or Integers.) More generally, FindRoot[eqns,{{x,x0},{y,y0},...}] numerically solves the list of equations/inequalities eqns for the list of variables {x,y,...} starting from {x0,y0,...}: In[2]:= FindRoot[Cos[x] == x,{x,0}] Out[2]= {x -> }

58 Mathematica 7.0.1: Solving Equations Solving differential equations: DSolve[eqns,{y1[x],y2[x],...},x] solves the list of differential equations/inequalities eqns for the list of functions {y1[x],y2[x],...}: In[1]:= DSolve[y [x] == 1,y[x],x] Out[1]= {{y[x] -> x+c[1]}}

59 Mathematica 7.0.1: Solving Equations Solving differential equations: DSolve[eqns,{y1[x],y2[x],...},x] solves the list of differential equations/inequalities eqns for the list of functions {y1[x],y2[x],...}: In[1]:= DSolve[y [x] == 1,y[x],x] Out[1]= {{y[x] -> x+c[1]}} Including boundary conditions: In[2]:= y[x]/.dsolve[{y [x] == 1,y[1]==3},y[x],x][[1]] Out[2]= 2+x

60 Mathematica 7.0.1: Solving Equations Solving differential equations: NDSolve[eqns,{y1[x],y2[x],...},{x,xmin,xmax}] numerically solves the same system between xmin and xmax: In[3]:= NDSolve[{y [x]==sin[x],y[0]==1},y[x],{x,0,10}] Out[3]= {{y[x]->interpolatingfunction[{{0.,10.}},<>][x]}} Boundary/initial conditions must be provided in this case.

61 Mathematica 7.0.1: Plotting Basic plot: Plot[Exp[-x],{x,0,5}]

62 Mathematica 7.0.1: Plotting Parametric plot: ParametricPlot[{2 Cos[t],Sin[t]},{t,0,2Pi}]

63 Mathematica 7.0.1: Plotting Contour plot: ContourPlot[Sin[x y],{x,-pi,pi},{y,-pi,pi}]

64 Mathematica 7.0.1: Plotting Contour plot with specific curves: ContourPlot[Sin[x y] == Range[0,1,0.1],{x,-Pi,Pi},{y,-Pi,Pi}]

65 Mathematica 7.0.1: Plotting 3D plot: Plot3D[Sin[x y],{x,-pi,pi},{y,-pi,pi}] Warning: high quality!

66 Mathematica 7.0.1: Plotting (Advanced) List plot: (syntax is slightly more complex) ListPlot[{{x1,y1},{x2,y2},...}] plots the points (x1,y1), (x2,y2) etc.

67 Mathematica 7.0.1: Plotting (Advanced) List plot: (syntax is slightly more complex) ListPlot[{{x1,y1},{x2,y2},...}] plots the points (x1,y1), (x2,y2) etc. Suitable lists are usually either: 1. Generated using Table[...] 2. Constructed from separate lists {x1,x2,...} and {y1,y2,...}.

68 Mathematica 7.0.1: Plotting (Advanced) List plot: (syntax is slightly more complex) ListPlot[{{x1,y1},{x2,y2},...}] plots the points (x1,y1), (x2,y2) etc. Suitable lists are usually either: 1. Generated using Table[...] 2. Constructed from separate lists {x1,x2,...} and {y1,y2,...}. Table[...] is an extension of Range[...]: In[1]:= Table[x^2,{x,1,9,2}] Out[1]= {1,9,25,49,81} Commands like Table[{f[x],g[x]},{x,0,2 Pi,0.1}] create lists of points suitable for ListPlot[...].

69 Mathematica 7.0.1: Plotting (Advanced) List plot (with Table): ListPlot[Table[{2 Cos[t],Sin[t]},{t,0,2 Pi,0.1}]]

70 Mathematica 7.0.1: Plotting (Advanced) List plot (with Transpose): Suppose instead we wish to plot ydata={y1,y2,...} against xdata={x1,x2,...} for given lists. We can mesh these separate lists together correctly using Transpose[...]: ListPlot[Transpose[{xdata,ydata}]] will produce the desired result. From the 2d Graphics Tips and Tricks sheet at

71 Mathematica 7.0.1: Plotting Plotting options: PlotRange->{{xmin,xmax},{ymin,ymax},{zmin,zmax}} AxesLabel->{"x-axis label","y-axis label"} PlotLabel->"plot label" PlotStyle->{Color,Linestyle,Linewidth} Axes->True/False Frame->True/False Joined->True (for ListPlot)

72 Mathematica 7.0.1: Plotting Overlaid plots (using lists): Plot[{Sin[x],Cos[x]},{x,0,2Pi}, PlotStyle->{{Red,Dashed},{Blue,Dotted}}]

73 Mathematica 7.0.1: Plotting Overlaid plots (using Show[...]): plot1 = ParametricPlot[{Cos[t]+0.1Cos[20t],Sin[t]+ 0.1Sin[20t]},{t,0,2Pi}]; plot2 = ListPlot[Table[{Cos[t],Sin[t]}, {t,0,2pi,pi/4}],plotstyle->{red}, Joined->{True}]; Show[plot1,plot2]

74 Mathematica 7.0.1: Remote Access Remote access: Command line: connect using ssh -p then type math to run Mathematica. This mode is interactive only (no notebook-style formatting). Windowed (slow): connect using ssh -p Y then type Mathematica. The full Mathematica GUI is displayed in this mode.

75 Mathematica 7.0.1: Other Tricks To clear a definition, use Clear[...] To clear all definitions, use ClearAll["Global *"]. (It is useful to place this line in the first input cell of every notebook.)

76 Mathematica 7.0.1: Other Tricks To clear a definition, use Clear[...] To clear all definitions, use ClearAll["Global *"]. (It is useful to place this line in the first input cell of every notebook.) To flatten nested lists, use Flatten[...]

77 Mathematica 7.0.1: Other Tricks To clear a definition, use Clear[...] To clear all definitions, use ClearAll["Global *"]. (It is useful to place this line in the first input cell of every notebook.) To flatten nested lists, use Flatten[...] It is possible to combine some commands using a piping (aka postfix) structure. Commands are stacked using the // operator: In[1]:= Pi/2 // N 1/Sqrt[2] // ArcSin // N Out[1]= Out[2]= This works for most single-argument functions (FullSimplify, TeXForm etc.)

78 Mathematica 7.0.1: Other Neat Features Scientific data sets for e.g. weather, planet positions are built in an can be accessed using functions like WeatherData[...], AstronomicalData[...], etc.

79 Mathematica 7.0.1: Other Neat Features Scientific data sets for e.g. weather, planet positions are built in an can be accessed using functions like WeatherData[...], AstronomicalData[...], etc. The generic Graphics[...] environment can be used to create decent-looking diagrams.

80 Mathematica 7.0.1: Other Neat Features Scientific data sets for e.g. weather, planet positions are built in an can be accessed using functions like WeatherData[...], AstronomicalData[...], etc. The generic Graphics[...] environment can be used to create decent-looking diagrams. Mathematica can produce animations and manipulable graphics using Animate[...] and Manipulate[...] in combination with Plot and Graphics.

81 Mathematica 7.0.1: Other Neat Features Scientific data sets for e.g. weather, planet positions are built in an can be accessed using functions like WeatherData[...], AstronomicalData[...], etc. The generic Graphics[...] environment can be used to create decent-looking diagrams. Mathematica can produce animations and manipulable graphics using Animate[...] and Manipulate[...] in combination with Plot and Graphics. See accompanying notebook for examples of these.

82 Mathematica 7.0.1: Quirks Quirks to watch out for: Undo button exists but rarely functions in any useful way.

83 Mathematica 7.0.1: Quirks Quirks to watch out for: Undo button exists but rarely functions in any useful way. Highlighting with arrow keys is double ended :-S

84 Mathematica 7.0.1: Quirks Quirks to watch out for: Undo button exists but rarely functions in any useful way. Highlighting with arrow keys is double ended :-S Mathematica always seems to try algebraic manipulation first. This can lead to problems when plotting functions with additional (numerical) parameters. In this case, using a?numericq pattern typically avoids the issue.

85 Mathematica 7.0.1: Quirks Quirks to watch out for: Undo button exists but rarely functions in any useful way. Highlighting with arrow keys is double ended :-S Mathematica always seems to try algebraic manipulation first. This can lead to problems when plotting functions with additional (numerical) parameters. In this case, using a?numericq pattern typically avoids the issue. There is no equivalent of Matlab s workspace - no easy way to check which variables are already defined.