Mathematica Cookbook

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Mathematica Cookbook helps you master the application's core principles by walking you through real-world problems. Ideal for browsing, this book includes recipes for working with numerics, data structures, algebraic equations, calculus, and statistics. You'll also venture into exotic territory with recipes for data visualization using 2D and 3D graphic tools, image processing, and music.

Although Mathematica 7 is a highly advanced computational platform, the recipes in this book make it accessible to everyone -- whether you're working on high school algebra, simple graphs, PhD-level computation, financial analysis, or advanced engineering models.

• Learn how to use Mathematica at a higher level with functional programming and pattern matching
• Delve into the rich library of functions for string and structured text manipulation
• Learn how to apply the tools to physics and engineering problems
• Draw on Mathematica's access to physics, chemistry, and biology data
• Get techniques for solving equations in computational finance
• Learn how to use Mathematica for sophisticated image processing
• Process music and audio as musical notes, analog waveforms, or digital sound samples
  

Title:Mathematica CookbookBy:Sal ManganoPublisher:O'Reilly MediaFormats:

• Print
• Ebook
• Safari Books Online
  

Print:May 2010Ebook:April 2010Pages:830Print ISBN:978-0-596-52099-1| ISBN 10:0-596-52099-9Ebook ISBN:978-1-4493-8324-4| ISBN 10:1-4493-8324-6

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关键词：mathematica Mathematic Thematic Cookbook Mathe

Problem

You want numerical results that are to a specified numerical precision and accuracy.

1. Solution
   
2. Use N[] to convert from exact to approximate form while controlling precision and accuracy to the desired amount.
   
3. 
   
4. In[19]:= N[1/5]
   
5. Out[19]= 0.2
   
6. You can explicitly specify the precision as a second argument to N[].
   
7. 
   
8. In[20]:= N[1/17,10]
   
9. Out[20]= 0.05882352941
   
10. You can also explicitly specify both the precision and accuracy, but this is less common. You might do this to guarantee a fixed number of decimal places independent of the size of the actual number.
    
11. 
    
12. In[21]:= N[{17, 1/17}, {Infinity, 10}]
    
13. Out[21]= {17.000000000, 0.0588235294}
    
14. To drive this point home, I ask you to consider the following. The first column uses fixed precision, whereas the second uses infinite precision and fixed accuracy.
    
15. 
    
16.      In[22]:= Table[With [{x = 10^n + 1 /17}, {N[x, 10], N[x, {Infinity, 10}]}],
    
17.                {n, 0, 5}] // TableForm

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Problem

You need to predict what Mathematica will do with expressions containing mixed types and representations.

1. Solution
   
2. The general rule of thumb is that the least precise type will determine the type of the result.
   
3. 
   
4. Mixing exact values and symbols
   
5. When expressions containing exact numeric values (integers and rationals) are mixed with symbols, Mathematica will keep all results in the most general form, possibly reducing rationals to integers but leaving symbolic values in symbolic form.
   
6. 
   
7. Mixing exact values and symbols
   
8. Mixing exact values and approximate values
   
9. When an approximate value is used in an otherwise symbolic expression, it forces Mathematica to convert to approximate values.
   
10. 
    
11. In[48]:= (2.0 Pi) / 3 + Pi / 3
    
12. Out[48]= 3.14159
    
13. 
    
14. In[49]:= 1. + (2 Pi) / 3 + Pi / 3
    
15. Out[49]= 4.14159
    
16. Mixing values of different precision and accuracy
    
17. When you mix values of different precision and accuracy, the lower precision and accuracy will determine the result. For multiplication, the precision of the result will be exactly the minimum of the precision of each term, whereas the accuracy will be somewhat less.
    
18. 
    
19. In[50]:= x = N[Sqrt[2], 30] * N[Sqrt[3], 10]
    
20. Out[50]= 2.449489743
    
21. 
    
22. In[51]:= Precision[x]
    
23. Out[51]= 10.
    
24. 
    
25. In[52]:= Accuracy[x]
    
26. Out[52]= 9.61092
    
27. For addition, the accuracy of the result will be exactly the minimum of the accuracy of each term; the precision will be somewhat more.
    
28. 
    
29. In[53]:= x = N[Sqrt[5], {Infinity, 30}] + N[Sqrt[7], {Infinity, 10}]
    
30. Out[53]= 4.8818192886
    
31. 
    
32. In[54]:= Precision[x]
    
33. Out[54]= 10.6886
    
34. 
    
35. In[55]:= Accuracy[x]
    
36. Out[55]= 10.

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