daviddoji/CalculusWithJuliaNotes.jl
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jverzani
2022-07-24 16:38:24 -04:00
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CwJ/precalc/polynomials_package.jmd
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@@ -187,11 +187,8 @@ observed from the output. The lone real root is approximately
irrational root.
```julia;echo=false
note("""
`SymPy` also has a `roots` function. If both `Polynomials` and `SymPy` are used together, calling `roots` must be qualified, as with `Polynomials.roots(...)`. Similarly, `degree` is provided in both, so it too must be qualified.
""")
```
!!! note
`SymPy` also has a `roots` function. If both `Polynomials` and `SymPy` are used together, calling `roots` must be qualified, as with `Polynomials.roots(...)`. Similarly, `degree` is provided in both, so it too must be qualified.
The `roots` function numerically identifies roots. As such, it is susceptible to floating point issues. For example, the following polynomial has one root with multiplicity ``5``, but ``5`` distinct roots are numerically identified:
@@ -352,8 +349,8 @@ Mathematically we say the ``0`` polynomial has no degree. What convention does `
```julia; hold=true; echo=false;
choices = ["`nothing`", "`-1`", "`0`", "`Inf`", "`-Inf`"]
ans = 2
radioq(choices, ans, keep_order=true)
answ = 2
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -511,10 +508,7 @@ It is ``0\cdot T_1(x) + 1\cdot T_1(x) + 2\cdot T_2(x) + 3\cdot T_3(x) = -2 - 8\c
radioq(choices, 3)
```
```julia; hold=true; echo=false
note("""
The `Polynomials` package has an implementation, so you can check your answer through `convert(Polynomial, ChebyshevT([0,1,2,3]))`. Similarly, the `SpecialPolynomials` package has these and many other polynomial bases represented.
!!! note
The `Polynomials` package has an implementation, so you can check your answer through `convert(Polynomial, ChebyshevT([0,1,2,3]))`. Similarly, the `SpecialPolynomials` package has these and many other polynomial bases represented.
The `ApproxFun` package is built on top of polynomials expressed in this basis, as the Chebyshev polynomials have special properties which make them very suitable when approximating functions with polynomials. The `ApproxFun` package uses easier-to-manipulate polynomials to approximate functions very accurately, thereby being useful for investigating properties of non-linear functions leveraging properties for polynomials.
""")
```
The `ApproxFun` package is built on top of polynomials expressed in this basis, as the Chebyshev polynomials have special properties which make them very suitable when approximating functions with polynomials. The `ApproxFun` package uses easier-to-manipulate polynomials to approximate functions very accurately, thereby being useful for investigating properties of non-linear functions leveraging properties for polynomials.