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Notebook[{

Cell[CellGroupData[{
Cell["Symbolic Computation", "Section"],

Cell[TextData[{
  "This notebook is designed to accompany Chapter 6 of Introduction to \
Scientific Programming: Computational Problem Solving Using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and C by Joseph L. Zachary.  In it we will explore the idea of symblic \
computing further."
}], "Text"],

Cell[CellGroupData[{

Cell["Introduction", "Subsection"],

Cell[TextData[{
  " Chapter 6 only hints at the support for symbolic mathematics provided by \
",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  In this notebook we will explore this support in more depth.  The \
material that we will be covering here requires varying levels of \
mathematical expertise.  Feel free to skip over examples if you don't have \
the necessary background."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Algebraic simplification", "Subsection"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s \"Simplify\" function expects an expression as a parameter, which it \
simplifies.  For example, notice the difference between"
}], "Text"],

Cell[BoxData[
    \(Sin[x]^2\  + \ Cos[x]^2\)], "Input"],

Cell["and", "Text"],

Cell[BoxData[
    \(Simplify[Sin[x]^2\  + \ Cos[x]^2]\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Equation solving", "Subsection"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " provides two general-purpose functions for solving equations, Solve and \
FindRoot.  Solve works symbolically much as you do when solving equations; \
FindRoot works numerically by repeatedly guessing at a solution until it \
finds one that works.  Solve is exact, FindRoot is approximate.  For \
complicated equations, Solve can be much slower than FindRoot.  In fact, \
there are plenty of equations where Solve fails but FindRoot succeeds."
}], "Text"],

Cell["\<\
Solve knows all kinds of tricks for solving equations.  For \
example, it knows the quadratic formula\
\>", "Text"],

Cell[BoxData[
    \(Solve[3*x^2\  - \ 10*x\  + \ 6\  == \ 0, \ x]\)], "Input"],

Cell["\<\
With floating-point coefficients, we get floating-point \
solutions.\
\>", "Text"],

Cell[BoxData[
    \(Solve[3.*x^2\  - \ 10.*x\  + \ 6.\  == \ 0, \ x]\)], "Input"],

Cell["Complex roots are also possible", "Text"],

Cell[BoxData[
    \(Solve[3*x^2\  - \ 4*x\  + \ 6\  == \ 0, \ x]\)], "Input"],

Cell["The coefficients can be entirely symbolic.", "Text"],

Cell[BoxData[
    \(Solve[a*x^2\  + \ b*x\  + \ c\  == \ 0, \ x]\)], "Input"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can also solve simultaneous linear equations.  Notice how list notation \
is used here."
}], "Text"],

Cell[BoxData[
    \(Solve[{3*x\  + \ 4*y\  == \ 7, \ 5*x\  + \ 3*y\  == \ 11}, \ {x, y}]\)],
   "Input"],

Cell["The equations don't have to be linear.", "Text"],

Cell[BoxData[
    \(Solve[{3*x^2\  + \ y\  == \ 7, \ 5*x\  + \ 3*y\  == \ 11}, \ {x, y}]\)],
   "Input"],

Cell["\<\
It is not difficult to pose an equation that solve cannot deal \
with.\
\>", "Text"],

Cell[BoxData[
    \(Solve[3*Cos[x] == \ x, \ x]\)], "Input"],

Cell["\<\
If you don't need an exact or a symbolic solution, FindRoot can \
solve more equations and is generally faster.\
\>", "Text"],

Cell[BoxData[
    \(FindRoot[3*Cos[x]\  == \ x, \ {x, 0}]\)], "Input"],

Cell["Of course, if you forget and pose a symbolic equation", "Text"],

Cell[BoxData[
    \(FindRoot[a*cos[x]\  == \ x, \ {x, 0}]\)], "Input"],

Cell["you'll get an error message.", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Calculus", "Subsection"],

Cell["\<\
Mathematica is also handy for doing calculus.  The D function does \
symbolic differentiation\
\>", "Text"],

Cell[BoxData[
    \(D[Cos[Sin[x]], \ x]\)], "Input"],

Cell["whereas the Integrate function does symbolic integration", "Text"],

Cell[BoxData[
    \(Integrate[\(-Sin[Sin[x]]\)*Cos[x], \ x]\)], "Input"],

Cell["You can also do definite integration by providing bounds", "Text"],

Cell[BoxData[
    \(Integrate[\(-Sin[Sin[x]]\)*Cos[x], \ {x, 0, Pi/2.}]\)], "Input"],

Cell["It is not difficult to stump Mathematica.", "Text"],

Cell[BoxData[
    \(Integrate[Cos[Sin[x]], \ x]\)], "Input"],

Cell["\<\
and this is how it tells you that it has failed.  In situations \
like this, trying to do a definite integral as above may be futile as well.\
\
\>", "Text"],

Cell[BoxData[
    \(Integrate[Cos[Sin[x]], \ {x, 0, Pi/2.}]\)], "Input"],

Cell[TextData[{
  "In this case, if you \"NIntegrate\" instead, ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will do a numerical integration.  This means that instead of first \
integration symbolically and then plugging in the bounds, it will find an \
approximation to the integral without first computing the definite integral."
}], "Text"],

Cell[BoxData[
    \(NIntegrate[Cos[Sin[x]], \ {x, 0, Pi/2.}]\)], "Input"],

Cell["\<\
It is possible to supply infinity and -infinity as bounds.  \
Thus\
\>", "Text"],

Cell[BoxData[
    \(Integrate[1/x, {x, 0, Infinity}]\)], "Input"],

Cell["diverges, whereas", "Text"],

Cell[BoxData[
    \(Integrate[1/Exp[x], \ {x, 0, Infinity}]\)], "Input"],

Cell["gives us a finite answer.", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Replacements", "Subsection"],

Cell[TextData[{
  "Another handy feature of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is the ability to perform substitutions.  Let's suppose that we solve an \
equation numerically"
}], "Text"],

Cell[BoxData[
    \(soln\  = \ FindRoot[Cos[x] == x, \ {x, 0}]\)], "Input"],

Cell["\<\
We can substitute the value of soln for x in the left-hand-side of \
the equation by doing\
\>", "Text"],

Cell[BoxData[
    \(ReplaceAll[Cos[x], \ soln]\)], "Input"],

Cell["\<\
This works for any kind of expression.  For example, suppose we \
differentiate\
\>", "Text"],

Cell[BoxData[
    \(res\  = \ D[Exp[Cos[x]], \ x]\)], "Input"],

Cell["\<\
and we'd like to find out the value of res when x is 1.2.  We can \
do that with\
\>", "Text"],

Cell[BoxData[
    \(ReplaceAll[res, \ {x -> 1.2}]\)], "Input"],

Cell["\<\
When an equation has multiple solutions, it is possible to pick \
them out individually.  Thus\
\>", "Text"],

Cell[BoxData[
    \(solns\  = \ Solve[3.15*x^2\  - \ 24.2*x\  + \ 15\  == \ 0, \ x]\)], 
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Cell["We can access the first of the two solutions as", "Text"],

Cell[BoxData[
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Cell["and the second of the two solutions as", "Text"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["Vectors and matrices", "Subsection"],

Cell[TextData[{
  "Vectors and matrices are used extensively in linear algebra.  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will do all of the standard operations on them."
}], "Text"],

Cell["Now we can define a vector with", "Text"],

Cell[BoxData[
    \(v\  = \ {1, 2, 3}\)], "Input"],

Cell[TextData[{
  "We can ask ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to display the vector using more standard notation by using the function \
\"MatrixForm\"."
}], "Text"],

Cell[BoxData[
    \(MatrixForm[v]\)], "Input"],

Cell["We can define a 3x3 matrix with", "Text"],

Cell[BoxData[
    \(m\  = \ {{1, 2, 3}, {4, 5, 4}, {3, 2, 1}}\)], "Input"],

Cell["and display it with", "Text"],

Cell[BoxData[
    \(MatrixForm[m]\)], "Input"],

Cell["\<\
Let's do some operations on vectors and matrices.  We can do \
addition,\
\>", "Text"],

Cell[BoxData[{
    \(MatrixForm[v + v]\), 
    \(MatrixForm[m + m]\)}], "Input"],

Cell["matrix multiplication,", "Text"],

Cell[BoxData[{
    \(MatrixForm[m.m]\), 
    \(MatrixForm[m.v]\)}], "Input"],

Cell["and dot product", "Text"],

Cell[BoxData[
    \(MatrixForm[v\ .\ v]\)], "Input"],

Cell["We can transpose", "Text"],

Cell[BoxData[
    \(MatrixForm[Transpose[m]]\)], "Input"],

Cell["or invert", "Text"],

Cell[BoxData[{
    \(minverse\  = \ Inverse[m]\), 
    \(MatrixForm[m\ .\ minverse]\)}], "Input"],

Cell["\<\
Much more is possible. Consult the on-line documentation for \
pointers.\
\>", "Text"]
}, Closed]]
}, Open  ]]
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