2.7.6 conjugate gradient method for a sparse system shi & bo
TRANSCRIPT
![Page 1: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/1.jpg)
2.7.6Conjugate Gradient Method for a Sparse SystemShi & Bo
![Page 2: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/2.jpg)
What is sparse system• A system of linear equations is called sparse if only a
relatively small number of its matrix elements are nonzero. It is wasteful to use general methods of linear algebra on such problems, because most of the O() arithmetic operations devoted to solving the set of equations or inverting the matrix involve zero operands. Furthermore, you might wish to work problems so large as to tax your available memory space, and it is wasteful to reserve storage for unfruitful zero elements. Note that there are two distinct (and not always compatible) goals for any sparse matrix method: saving time and/or saving space.
![Page 3: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/3.jpg)
Conjugate Gradient Method
Ax=b
The ordinary conjugate gradient algorithm:▫steepest descent method
![Page 4: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/4.jpg)
steepest descent method
•Where A is n*n symmetric positive definite matrix, b is known n-d vector.
•Solving Ax=b is equivalent to minimizing
•▽f=Ax-b
![Page 5: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/5.jpg)
steepest descent method
•Steepest Descent▫Start at a initial guess point adjust until
close enough to the exact solution:
▫Where i is number of iterations is step size, is Adjustment Direction.
•How to choose direction and step size?
![Page 6: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/6.jpg)
Choose direction
Choose the direction which f decreases most quickly. Move from point to the point by minimizing along the line from in the direction opposite to .
Hense, .
![Page 7: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/7.jpg)
Choose step size
Step size should minimize f, along the direction of ,which means
![Page 8: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/8.jpg)
When to stop
We should give a stopping criterion because there may have many errors and noises.
(1)only for exact arithmetic, not in practice(2) in practice unstable alg for general A
stopping criterion is or with an given small .
![Page 9: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/9.jpg)
Symmetric but non-positive definite A
•With the choice instead of . In this case and for all k. This algorithm is equivalent to the ordinary conjugate gradient algorithm, but with all dot products replaced by . It is called the minimum residual algorithm, because it corresponds to successive minimizations of the function
![Page 10: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/10.jpg)
For any nonsingular matrix A, is symmetric and positive-definite.
But we can’t use .
Because the condition number of the matrix is the square of the condition number of A.
![Page 11: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/11.jpg)
•Thanks
•Reference▫Numerical Recipes▫Steepest Decent and Conjugate Gradients,
w3.pppl.gov/m3d/reference/SteepestDecentandCG.ppt
![Page 12: 2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d1e5503460f949f1d31/html5/thumbnails/12.jpg)
Reference
•Numerical Recipes•Steepest Decent and Conjugate Gradients