Le of a sparse matrix the procedure begins with a

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Unformatted text preview: aad's (1996) 25 Cuthill-McKee Algorithm Example 11. Consider the mesh shown 2 1 1 0 1 0 1 0 1 0 1 2 1 0 1 0 4 XX 5 1 04 1 0 X X 6 X X XXX X 6 1 0 1 0 5 XX 3 4 X 2 3 1 3 XXX 1 0 1 0 5 6 Cuthill-McKee procedure with starting vertex 3 Level Set Vertex Neighbor Degree Q 1 3 1 4 f3, 5 4 1, 5, 2 1 2 3 4 4 2, 4, 5 6 3 6g New numbering 2 4 1 0 1 0 1 0 1 0 1 0 1 2 3 4 5 XXX 4 X 5 1 05 1 0 XX 3 1 0 1 0 X 2 1 1 XXXX 6 1 0 1 0 3 1 0 1 0 6 26 6 X XXXX Cuthill-McKee Algorithm Example 12. Consider the mesh and graph on the next page { Unknowns at midpoint nodes are only connected to those on elements containing the node { e.g., quadratic nite element approximation With row-by-row ordering: p = 13, P = 106 First level set { Start at Node 7 Q = f7g { Traverse the rst level set Vertex Neighbors Degree Q 7 1 11 f7, 4 5 8 8 11 5 14 11 4, 11, 8, 1, 14g { After the rst level-set traversal: L = f4 11 8 1 14g Q = f7 4 11 8 1 14g 27 Example 12: Row-by-Row Order 1 1 0 1 0 1 1 02 1 0 2 2 111 0 0 03 111 000 5 4 6 5 6 40 0000 1 1111 1 1111 0 0000 7 1 0 1 0 8 111 0 9 0 010 111 000 7 9 8 1 0 1 0 10 1 0 1 0 13 1 1111 0 0000 1 1111 0 0000 11 12 111 000 111 000 14 15 12 11 16 15 1 0 1 140 12 13 34 5 6 7 8 9 10 11 12 13 14 15 1 016 1 0 16 1X 2X X 3X X X 4X X 5X X X X 6X X X XX 7X X X 8X XX XX 9X X X XX X X 10 11 X X 12 X 13 XX XX X X 14 X XX XX X XX XX XXX X XX XXXX X X X 28 XXXX XX 15 16 X XXXXX Example 12: Cuthill-McKee Order Level Set 2: L = f4 11 8 1 14g Q = f7 4 11 8 1 14g Vertex Neighbors Degee 4 11 8 5 8 9 12 12 8 1 2 5 3 8 6 8 14 13 8 15 5 16 8 { The new order L = f5 Q = f7 12 9 2 3 6 15 13g 4 11 8 1 14 5 12 9 2 3 6 15 13 16g Third level set: The only unnumbered vertex is 10 Q = f7 4 11 8 1 14 5 12 9 2 3 6 15 13 16 10g 29 Example 12: Cuthill-McKee Order Bandwidth and pro le: p = 9, P = 76 5 10 11 111 000 111 000 2 7 12 1 1111 0 0000 1 1111 0 0000 10 1 1 0 4 111 0 9 016 0 111 000 8 1 1111 0 0000 1 1111 0 0000 14 3 111 000 111 000 6 12 34 5 6 13 7 8 15 9 10 11 12 13 14 15 16 1X 2X X 3X X X 4X X X X 5X X X X X 6X X X X X X 7 XXXX 8 XXXXX 9 XXXXXX 10 X X XX 11 X X XXX 12 X X XXXX 13 X XX 14 X XX XXXX 15 X XX XXXXX X 16 30 X XX XXX eeKcM-llihtuC Algorithm George (1971) 1 reversed the numbering given by the CuthillMcKee algorithm { This is called the Reverse Cuthill-McKee (RCM) algorithm { George proves that the pro le of RCM is never larger than that of CM The RCM ordering for Example 12 is f10 16 13 15 6 3 2 9 12 5 14 1 8 11 4 7g The MATLAB procedure symrcm performs RCM on symmetric matrics { It produced the same ordering as above with 12 and 5 interchanged and 4 and 7 interchanged { Evidently, 4 was the starting node { This gave the same pro le as the above ordering The MATLAB procedure spy will plot the nonzero structure of a sparse matrix J.A. George (1971), \Computer Implementation of the Finite Element Method," Rep. STAN-CS71-208, Dept. Comp. Sci., Stanford University, Palo Alto 1 31 Example 12: RCM Oder Bandwidth and pro le: p = 9, P = 68 12 7 6 111 000 111 000 10 5 15 0 0000 1 1111 1 1111 0 0000 11 16 0 13 0 8 11 00 11 0 01 11 00 9 1 1111 0 0000 1 1111 140 0000 3 111 0 0 02 111 000 11 12 34 5 6 4 7 8 9 10 11 12 13 14 15 16 1X 2X X 3X X X 4 XXX 5X X X X 6X X X XX 7 XXX 8X X X X X 9 12 13 XX XXX XXXXXX 10 11 XXX XXXX XXX X XXXXXXX XXXXXX 14 XXXX 15 XXXXX 16 XXXXXX 32 Wave-Front Ordering \Frontal" or \wave front" techniques: { Select unknowns xi that only depend on themselves They form the initial level set If none, select an unknown on the periphery of the graph { All unknowns that only involve those in Level Set 1 form Level Set 2 { All unknowns that only involve those in Level Sets 1 and 2 form Level Set 3, etc. For nite element problems, this can be regarded as element annihilation { cf. J.A. George and J.W. Liu (1981) Example 13. Consider the problem of Example 12 { Select an element and number the unknowns that are not connected to unknowns on other elements { Remove the element { Proceed to an adjacent element and repeat the process 33 11 00 11 00 Wave-Front Ordering 11 00 1 2 11 11 00 00 11 00 11 00 4 3 11 11 11 00 00 00 11 11 00 00 11 00 11 00 11 00 5 11 00 7 11 11 00 00 11 00 11 00 11 00 8 9 11 00 11 00 11 11 00 00 11 00 11 00 11 00 10 34 6 11 11 11 00 00 11 11 11 00 00 00 12 14 15 13 16 Wave-Front Ordering Wave-Front bandwidth and pro le: p = 13, P = 68 5 6 8 111 000 111 000 7 1 9 1 1111 0 0000 1 1111 0 0000 2 1 0 1 0 4 111 0 011 010 111 000 13 1 1111 3 0 0000 1 1111 0 0000 12 111 000 111 000 15 14 12 34 5 6 7 8 16 9 10 11 12 13 14 15 16 1X 2X X 3X X X 4X X X X 5X X X XX 6 XX 7 XXXX 8 X 9 XXXXX XXX XXX 10 11 XXXXXXXX 12 XX XX 13 14 X X X X X XX X XXXX X XXXXX 15 16 XXXX XXXXXXXXX 35...
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This document was uploaded on 03/16/2014 for the course CSCI 6800 at Rensselaer Polytechnic Institute.

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