CS475/CM375 Lecture 4: Sept 22 Sparse Gaussian Elimination, Graph - - PDF document

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21/09/2011 CS475/CM375 Lecture 4: Sept 22 Sparse Gaussian Elimination, Graph Representation Reading: [Saad] Sect 3.1 3.2 CS475/CM375 (c) 2011 P. Poupart & J. Wan 1 5 Point Stencil An easy way to denote 2D finite difference

SLIDE 1

21/09/2011 1

CS475/CM375 Lecture 4: Sept 22

Sparse Gaussian Elimination, Graph Representation Reading: [Saad] Sect 3.1‐3.2

5‐Point Stencil

An easy way to denote 2D finite difference equations
,

, , , ,

SLIDE 2

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Numbering of unknowns

Picture:
Note: the values on the boundary are zero
The unknowns are:
,

, ⋯ ,

, ⋯ , ⋮ ⋮ ⋱ ⋮

, ⋯

,
Total number ≡

Natural ordering

Ordering: first in the x‐direction, then y‐direction

– i.e.,

,, ,, … , ,; ,, ,, …

The system of linear equations

1, 1:

2, 1:

⋮ , :

SLIDE 3

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Matrix Form

Example 4, 16

Graph Representation of Matrices

Given a sparse matrix , a node is associated with

each row.

If , 0, there exists an edge from node to
1

2 3 4 :

SLIDE 4

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Graph for Symmetric Matrices

For symmetric matrices, arrows can be dropped (as

well as self loops)

2 3 4 :

Physical/Geometric Interpretation

Graph of a matrix often has a simple

physical/geometric interpretation

– 1D Laplacian : – 2D Laplacian :

SLIDE 5

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GE and Matrix Graph

“Visualize” eliminations by matrix graph

e.g.

GE fill‐in

GE and Matrix Graph

Elimination of node produces a new graph with

– Node deleted, all edges containing node deleted – New edge , added (fill‐in) if there was an edge , & , in the old graph.

Notes

– Matrix (with symmetric structure) graph is unchanged by renumbering of the nodes – But orderings (which nodes to be removed first) may result in much less fill during GE.

SLIDE 6

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Ordering Algorithms

Consider the following matrix graph:
Assume ≫ . If we use the natural ordering,

what would the matrix look like?

CS475/CM375 (c) 2011 P. Poupart & J. Wan

Ordering Algorithms

If we had numbered along y‐direction first, the

matrix becomes:

Which ordering results in less fill? Why?

SLIDE 7

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Band Matrices

Note: GE preserves band structure

– Picture:

Amount of work to factor a band matrix:

– where bandwidth – x‐first ordering →

– y‐first ordering →
CS475/CM375 (c) 2011 P. Poupart & J. Wan

Envelope Methods

In general, bandwidth is not the same for each row

– Example:

In each row, fill can occur only between the 1st

nonzero entry and the diagonal.

To limit the amount of fill, keep the envelope as close

to the diagonal as possible

21/09/2011 1

CS475/CM375 Lecture 4: Sept 22

Sparse Gaussian Elimination, Graph Representation Reading: [Saad] Sect 3.1‐3.2

5‐Point Stencil

, , , ,

21/09/2011 2

Numbering of unknowns

, ⋯ ,

, ⋯ , ⋮ ⋮ ⋱ ⋮

, ⋯

Natural ordering

– i.e.,

1, 1:

2, 1:

⋮ , :

21/09/2011 3

Matrix Form

Graph Representation of Matrices

each row.

2 3 4 :

21/09/2011 4

Graph for Symmetric Matrices

well as self loops)

2 3 4 :

Physical/Geometric Interpretation

physical/geometric interpretation

– 1D Laplacian : – 2D Laplacian :

21/09/2011 5

GE and Matrix Graph

e.g.

GE fill‐in

GE and Matrix Graph

– Node deleted, all edges containing node deleted – New edge , added (fill‐in) if there was an edge , & , in the old graph.

– Matrix (with symmetric structure) graph is unchanged by renumbering of the nodes – But orderings (which nodes to be removed first) may result in much less fill during GE.

21/09/2011 6

Ordering Algorithms

what would the matrix look like?

Ordering Algorithms

matrix becomes:

21/09/2011 7

Band Matrices

– Picture:

– where bandwidth – x‐first ordering →

Envelope Methods

– Example:

nonzero entry and the diagonal.

to the diagonal as possible

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Envelope Methods

numbers as close together as possible

– Example: