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4 Methods with Poisson Matrix

4 Methods with Poisson Matrix. 7 조 이안준 이정훈 이지훈. Contents. Preliminary What is the ‘ poisson Matrix’? Eigenvalues of Poisson Matrix -Power Method -Inverse Power Method -QR Alg & Hessenberg QR Alg Summary Conclusion. What is the ‘Poisson Matrix’?.

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4 Methods with Poisson Matrix

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  1. 4 Methods withPoisson Matrix 7조 이안준 이정훈 이지훈

  2. Contents • Preliminary What is the ‘poisson Matrix’? • Eigenvalues of Poisson Matrix -Power Method -Inverse Power Method -QR Alg & Hessenberg QR Alg • Summary • Conclusion

  3. What is the ‘Poisson Matrix’?

  4. What is the ‘Poisson Matrix’?

  5. What is the ‘Poisson Matrix’? A=full(gallery( 'poisson’,3)) A = 4 -1 0 -1 0 0 0 0 0 -1 4 -1 0 -1 0 0 0 0 0 -1 4 0 0 -1 0 0 0 -1 0 0 4 -1 0 -1 0 0 0 -1 0 -1 4 -1 0 -1 0 0 0 -1 0 -1 4 0 0 -1 0 0 0 -1 0 0 4 -1 0 0 0 0 0 -1 0 -1 4 -1 0 0 0 0 0 -1 0 -1 4

  6. What is the ‘Poisson Matrix’? • A is block-tridiagonal • All diagonal elements of A are 4 • In each row, between 2 and 4 elements are non-zero(-1) • A is symmetric & diagonal dominant So, • All its eigenvalues are real • And it is also positive definite

  7. Calculating Eigenvalues • Power Method # of iteration = 25 E_value = 6.8284 E_vector = Dominance ratio = 0.7929 0.2500 -0.3536 0.2500 -0.3536 0.5000 -0.3536 0.2500 -0.3536 0.2500

  8. Calculating Eigenvalues • Inverse Power Method # of iteration = 9 E_value = 1.1716 E_vector = Dominance ratio = 0.4531 0.2500 0.3536 0.2500 0.3536 0.5000 0.3536 0.2500 0.3536 0.2500

  9. Calculating Eigenvalues • QR Algorithm # of iteration = 29 E_value = 6.8284 5.4142 4.0000 5.4142 4.0000 2.5858 4.0000 2.5858 1.1716

  10. Calculating Eigenvalues • Hessenberg QR Algorithm # of iteration = 30 E_value = 4.0000 5.4142 4.0000 2.5858 6.8284 5.4142 4.0000 2.5858 1.1716

  11. Summary

  12. Summary

  13. Conclusion • Convergence rate of PM & IPM

  14. Conclusion • Convergence rate of PM & IPM

  15. Conclusion • Why the difference between # of iteration of QR Alg. & Hess.QR Alg. is small?

  16. Conclusion • Why is the difference between # of iteration of QR Alg. & Hess.QR Alg. small? A = 4 -1 0 -1 0 0 0 0 0 -1 4 -1 0 -1 0 0 0 0 0 -1 4 0 0 -1 0 0 0 -1 0 0 4 -1 0 -1 0 0 0 -1 0 -1 4 -1 0 -1 0 0 0 -1 0 -1 4 0 0 -1 0 0 0 -1 0 0 4 -1 0 0 0 0 0 -1 0 -1 4 -1 0 0 0 0 0 -1 0 -1 4

  17. Conclusion • Why is the difference between # of iteration of QR Alg. & Hess.QR Alg. small? • After 1 iteration

  18. Conclusion • Why is the difference between # of iteration of QR Alg. & Hess.QR Alg. small? • After 10 iteration

  19. Conclusion • Why is the difference between # of iteration of QR Alg. & Hess.QR Alg. small? • After 20 iteration

  20. Q & A

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