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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"]
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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"],
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\(Solve[{3*x\ + \ 4*y\ == \ 7, \ 5*x\ + \ 3*y\ == \ 11}, \ {x, y}]\)],
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Cell["The equations don't have to be linear.", "Text"],
Cell[BoxData[
\(Solve[{3*x^2\ + \ y\ == \ 7, \ 5*x\ + \ 3*y\ == \ 11}, \ {x, y}]\)],
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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"]
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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"],
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\(Integrate[Cos[Sin[x]], \ {x, 0, Pi/2.}]\)], "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"],
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\(NIntegrate[Cos[Sin[x]], \ {x, 0, Pi/2.}]\)], "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"]
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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"],
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Cell["\<\
When an equation has multiple solutions, it is possible to pick \
them out individually. Thus\
\>", "Text"],
Cell[BoxData[
\(solns\ = \ Solve[3.15*x^2\ - \ 24.2*x\ + \ 15\ == \ 0, \ x]\)],
"Input"],
Cell["We can access the first of the two solutions as", "Text"],
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\(solns[\([1]\)]\)], "Input"],
Cell["and the second of the two solutions as", "Text"],
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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"],
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Cell[TextData[{
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StyleBox["Mathematica",
FontSlant->"Italic"],
" to display the vector using more standard notation by using the function \
\"MatrixForm\"."
}], "Text"],
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Cell["We can define a 3x3 matrix with", "Text"],
Cell[BoxData[
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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"]
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