Embed Size (px)
Citation preview
Newton’s Method: Homework• Create a Google Spreadsheet
– that uses Newton’s Method– to find roots of x^2 + x = 1
• Link to this page from your homepage• Send link(s) to:
– [email protected] and [email protected]• Extra credit:
– Multiple roots: There are two roots to this equation. You will get one root or the other depending on the initial value.• Show that this is the case. That is, produce two spreadsheets with two initial
values that converge on different roots.• Which initial values converge on which root?
– What happens if you try to use Newton’s Method to solve: x^2 + x = -1
Newton’s Methodhttp://archives.math.utk.edu/visual.calculus/3/newton.5/
1 3 5 7 9 11 13 152.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
sqrt(5)
Iteration (n)
Estim
ate
of s
qrt(
5)
n x[n] x[n+1] sqrt(5)1 3.000000 2.333333 2.2360682 2.333333 2.238095 2.2360683 2.238095 2.236069 2.2360684 2.236069 2.236068 2.2360685 2.236068 2.236068 2.2360686 2.236068 2.236068 2.2360687 2.236068 2.236068 2.2360688 2.236068 2.236068 2.2360689 2.236068 2.236068 2.236068
10 2.236068 2.236068 2.23606811 2.236068 2.236068 2.23606812 2.236068 2.236068 2.23606813 2.236068 2.236068 2.23606814 2.236068 2.236068 2.236068
Symbolic Features(Bet you can’t do this with your favorite statistics package)
• Complex Numbers: Sqrt(-1)• Roots (without approximations)• Differentiation (without approximations)• Integration (without approximations)• The On-Line Encyclopedia of Integer Sequences• Eval
• Symbolic Methods ≠ Numeric Methods– Newton’s Method: Approximation for Reals
Sqrt(-1) Error (for many tools)
Roots (without approximations)2/)51( x
Numerical Methods:Approximations such as Newton’s Method
Complex Roots
Newton’s Methodhttp://archives.math.utk.edu/visual.calculus/3/newton.5/
1 3 5 7 9 11 13 152.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
sqrt(5)
Iteration (n)
Estim
ate
of s
qrt(
5)
Newton’s Methodhttp://archives.math.utk.edu/visual.calculus/3/newton.5/
1 2 3 4 5 6 7 8 9 10 11 12 13 14 152.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
-5
0
5
10
15
20
sqrt(5) sqrt(-5)
Iteration (n)
Estim
ate
of s
qrt(
5)
Estim
ate
of s
qrt(
-5)
Symbolic Alternative
1 3 5 7 9 11 13 152.22.32.42.52.62.72.82.9
3
-5
0
5
10
15
20
sqrt(5) sqrt(-5)
Iteration (n)
Estim
ate
of s
qrt(
5)
Estim
ate
of s
qrt(
-5)
