math 175: numerical analysis ii lecturer: jomar fajardo rabajante 2 nd sem ay 2012-2013 imsp, uplb
TRANSCRIPT
![Page 1: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/1.jpg)
MATH 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante2nd Sem AY 2012-2013
IMSP, UPLB
![Page 2: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/2.jpg)
Numerical Methods for Linear Systems
Review (Naïve) Gaussian EliminationGiven n equations in n variables.
• Operation count for elimination step:(multiplications/divisions)
• Operation count for back substitution:
3
3nO
2
2nO
![Page 3: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/3.jpg)
Numerical Methods for Linear Systems
Overall (Naïve) Gaussian Elimination takes
Take note: we ignored here lower-order terms and we did not include row exchanges and additions/subtractions. WHAT MORE IF WE ADDED THESE STUFFS???!!! KAPOY NA!
323
323 nO
nO
nO
![Page 4: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/4.jpg)
Numerical Methods for Linear Systems
Example: Consider 10 equations in 10 unknowns. The approximate number of operations is
If our computations have round-off errors, how would our solution be affected by error magnification? Tsk… Tsk…
3343
103
![Page 5: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/5.jpg)
Numerical Methods for Linear Systems
Our goal now is to use methods that will efficiently solve our linear systems with minimized error
magnification.
![Page 6: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/6.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
• When we are processing column i in Gaussian elimination, the (i,i) position is called the pivot position, and the entry in it is called the pivot entry (or simply the pivot).
• Let [A|b] be an nx(n+1) augmented matrix.
![Page 7: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/7.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
STEPS:1. Begin loop (i = 1 to n–1):2. Find the largest entry (in absolute value) in
column i from row i to row n. If the largest value is zero, signal that a unique solution does not exist and stop.
3. If necessary, perform a row interchange to bring the value from step 2 into the pivot position (i,i).
![Page 8: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/8.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
4. For j = i+1 to n, perform
5. End loop.6. If the (n,n) entry is zero, signal that a unique
solution does not exist and stop. Otherwise, solve for the solution by back substitution.
jii,i
j,ij RRa
aR
![Page 9: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/9.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
Example:
8
12
1
1284
1244
221
8
1
12
1284
221
1244Original matrix (Matrix 0) Matrix 1
![Page 10: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/10.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
12
0
1244Matrix 1 Matrix 2
11
44 0
![Page 11: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/11.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
12
10
1244Matrix 1 Matrix 2
21
44 1
![Page 12: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/12.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
12
110
1244Matrix 1 Matrix 2
21
412 –1
![Page 13: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/13.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
2
12
110
1244Matrix 1 Matrix 2
11
412 –2
![Page 14: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/14.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
2
12
0
110
1244Matrix 1 Matrix 2
44
44 0
![Page 15: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/15.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
2
12
40
110
1244Matrix 1 Matrix 2
84
44 4
![Page 16: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/16.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
2
12
040
110
1244Matrix 1 Matrix 2
124
412 0
![Page 17: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/17.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
8
1
12
1284
221
1244
4
2
12
040
110
1244Matrix 1 Matrix 2
84
412 –4
![Page 18: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/18.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
2
4
12
110
040
1244
4
2
12
040
110
1244
Matrix 2 Matrix 3
![Page 19: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/19.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
2
4
12
110
040
1244
1
4
12
100
040
1244
Matrix 3 Final Matrix (Matrix 4)
![Page 20: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/20.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
1
4
12
100
040
1244
Final Matrix Back substitution:
1x
121244x
1212z4y4x
1y44y
1z1z
![Page 21: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/21.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
1
1
2
720
410
1290
8
2
12
000
1040
12191
a unique solution does not exist
![Page 22: MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante 2 nd Sem AY 2012-2013 IMSP, UPLB](https://reader036.vdocuments.net/reader036/viewer/2022062517/56649e925503460f94b97736/html5/thumbnails/22.jpg)
1st Method: Gaussian Elimination with Partial Pivoting
• There are other pivoting strategies such as the complete (or maximal) pivoting. But complete pivoting is computationally expensive.