newton’s method: homework create a google spreadsheet – that uses newton’s method – to find...
TRANSCRIPT
![Page 1: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/2.jpg)
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)
![Page 3: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/3.jpg)
![Page 4: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/6.jpg)
Sqrt(-1) Error (for many tools)
![Page 7: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/7.jpg)
Roots (without approximations)2/)51( x
![Page 8: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/8.jpg)
Numerical Methods:Approximations such as Newton’s Method
![Page 9: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/9.jpg)
Complex Roots
![Page 10: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/10.jpg)
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)
![Page 11: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/11.jpg)
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)
![Page 12: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/12.jpg)
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)
![Page 13: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/13.jpg)
![Page 14: 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](https://reader035.vdocuments.us/reader035/viewer/2022062716/56649dc95503460f94abfd0c/html5/thumbnails/14.jpg)