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Results for different matrices and comparisons Dense Matrices - PowerPoint PPT Presentation

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Results for different matrices and comparisons Dense Matrices Rectangular Matrices Rectangular Matrices Symmetric Matrices Sparse Matrices Dense Rectangular matrices Dense Rectangular matrices Dense Rectangular matrices Dense

  1. Results for different matrices and comparisons

  2. • Dense Matrices – Rectangular Matrices Rectangular Matrices – Symmetric Matrices • Sparse Matrices

  3. Dense Rectangular matrices Dense Rectangular matrices

  4. Dense Rectangular matrices Dense Rectangular matrices • Comparison of timings when m is fixed and n is varied n m Built in Comb. Comb. Indirect Direct Francis Built in Square Method Method SVD 100 0.0073 0.1779 0.5651 0.4488 0.1616 0.8897 200 0.0129 0.4382 1.6292 1.1313 0.4599 1.3890 700 700 0 0451 0.0451 11 4075 11.4075 25.4917 25 4917 14 3346 14.3346 11 6559 11.6559 3 6083 3.6083 100 800 0.0589 16.5906 35.1912 19.3266 16.7657 3.9636 1400 1.1683 76.9117 147.1903 71.8148 75.7396 7.6926 1500 1.3396 96.2016 181.0894 93.1726 94.5906 8.2684 100 0.0191 0.4518 1.6380 1.1398 0.4642 1.8997 200 0.0331 1.7899 3.5367 1.8215 1.9187 5.5780 700 0.1123 26.2992 40.5601 14.7202 26.1704 14.6077 200 800 0.1689 37.0253 55.9529 19.7481 37.0049 16.2683 1400 0.2952 162.2110 236.5831 74.3360 162.2961 27.0306 1500 0.3177 200.9886 290.3429 89.6555 201.3425 28.6722

  5. Dense Rectangular matrices Dense Rectangular matrices

  6. Dense Rectangular matrices Dense Rectangular matrices • Comparison of accuracies when n is fixed and m is varied Comb. Comb. Indirect Direct n m Built in Built in Square Method Method SVD 2.22e ‐ 13 9e ‐ 13 9.3e ‐ 13 5.5e ‐ 13 100 3.8e-13 200 5.66e ‐ 13 8.1e-13 8e ‐ 13 3e ‐ 13 13.7e ‐ 13 700 20.88e ‐ 13 30.04e ‐ 13 28e ‐ 13 10.2e ‐ 13 36.5e ‐ 13 100 25.78e ‐ 13 37.6e ‐ 13 33e ‐ 13 12.1e ‐ 13 46.7e ‐ 13 800 40.33e ‐ 13 67.6e ‐ 13 61e ‐ 13 25.1e ‐ 13 82.3e ‐ 13 1400 1500 47.24e ‐ 13 70.9e ‐ 13 60e ‐ 13 27.6e ‐ 13 84.1e ‐ 13 100 3.54e ‐ 13 5.6e ‐ 13 5e ‐ 13 1.6e ‐ 13 7.3e ‐ 13 5.69e 13 5 69e ‐ 13 11 3e ‐ 13 11.3e 13 1482e 13 1482e ‐ 13 247 6e ‐ 13 247.6e 13 18e ‐ 13 18e 13 200 200 20.56e ‐ 13 43.9e ‐ 13 43e ‐ 13 12.1e ‐ 13 52.6e ‐ 13 700 200 800 25.42e ‐ 13 50.8e ‐ 13 51e ‐ 13 14.3e ‐ 13 59.8e ‐ 13 1400 44.68e ‐ 13 98.3e ‐ 13 98e ‐ 13 27.6e ‐ 13 115.5e ‐ 13 48.22e ‐ 13 107.3e ‐ 13 102e ‐ 13 30.1e ‐ 13 129.3e ‐ 13 1500

  7. Dense Rectangular matrices Dense Rectangular matrices

  8. Dense Rectangular matrices Dense Rectangular matrices • Comparison of accuracies when m is fixed and n is varied n m Built in Comb. Comb. Indirect Direct Built in Square Method Method SVD 2.62 e-13 3.7 e-13 8.2 e-13 5.92 e-13 5.8 e-13 100 3.04 e-13 5.26 e-13 4.8 e-13 1.55 e-13 6.04 e-13 200 700 700 2 93 13 2.93 e-13 8.15 e-13 8 1 13 8.2 e-13 8 2 13 1 8 13 1.78 e-13 7.87 e-13 8 13 100 800 3.97 e-13 9.8 e-13 9.6 e-13 1.99 e-13 7.45 e-13 3.52 e-13 12.34 e-13 12 e-13 2.39 e-13 8.97 e-13 1400 3.2 e-13 17.57 e-13 17.1 e-13 2.31 e-13 10.59 e-13 1500 100 6.04 e-13 9.21 e-13 8.1 e-13 3.17 e-13 14.07 e-13 200 5.28 e-13 11.53 e-13 247.7 e-13 16.71 e-13 17.68 e-13 6.51 e-13 18.67 e-13 18.4 e-13 4.05 e-13 20.09 e-13 700 200 6.20 e-13 19.55 e-13 19.4 e-13 4.03 e-13 18.89 e-13 800 1400 5.86 e-13 21.5 e-13 21.5 e-13 4.38 e-13 21.15 e-13 1500 6.17 e-13 27.02 e-13 26.5 e-13 4.9 e-13 24.03 e-13

  9. Dense Rectangular matrices Dense Rectangular matrices

  10. Dense Rectangular matrices Dense Rectangular matrices

  11. Dense Rectangular matrices Dense Rectangular matrices

  12. Dense Rectangular matrices Dense Rectangular matrices • Orthogonality checks when m is fixed and n is varied m n Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id UU T ‐ Id VV T ‐ Id 0.28 e-13 0.22 e-13 0.03 e-13 0.77 e-13 0.71 e-13 100 0.19e-13 0.18e-13 0.28e-13 0.22e-13 0.03e-13 200 0.22 e-13 0.18 e-13 0.71 e-13 0.33 e-13 0.49 e-13 0.24 e-13 0.07 e-13 0.04 e-13 0.71 e-13 0.42 e-13 700 0.37 e-13 0.20 e-13 1.45 e-13 0.31 e-13 1.05 e-13 0.23 e-13 0.17 e-13 0.03 e-13 1.44 e-13 0.47 e-13 100 1.59 e-13 1.05 e-13 0.03 e-13 1.59 e-13 0.34 e-13 800 0.30 e-13 0.19 e-13 0.28 e-13 0.21 e-13 0.19 e-13 1 91 13 1.91 e-13 1.56 e-13 1 56 13 0 03 13 0.03 e-13 1.91 e-13 1 91 13 0.46 e-13 0 46 13 1400 1400 0.53 e-13 0 53 13 0 21 13 0.21 e-13 0 30 13 0.30 e-13 0.21 e-13 0 21 13 0.33 e-13 0 33 13 1500 0.51 e-13 0.19 e-13 1.79 e-13 0.31 e-13 1.82 e-13 0.22 e-13 0.35 e-13 0.03 e-13 1.79 e-13 0.37 e-13 100 0.20e-13 0.21e-13 0.31 e-13 0.70e-13 0.22 e-13 0.48e-13 0.04e-13 0.06 e-13 0.37 e-13 0.70 e-13 0.62 e-13 0.53 e-13 0.07 e-13 4.66 e-13 4.70 e-13 200 0.32 e-13 0.30 e-13 0.66 e-13 0.55 e-13 0.08 e-13 2.72 e-13 1.56 e-13 0.07 e-13 2.73 e-13 0.74 e-13 700 0.40 e-13 0.31 e-13 0.64 e-13 0.54 e-13 0.18 e-13 200 800 0.43 e-13 0.30 e-13 2.92 e-13 0.67 e-13 1.47 e-13 0.56 e-13 0.20 e-13 0.07 e-13 2.92 e-13 0.76 e-13 1400 0.57 e-13 0.30 e-13 3.06 e-13 0.74 e-13 1.97 e-13 0.76 e-13 0.32 e-13 0.06 e-13 3.05 e-13 0.81 e-13 3.56 e-13 2.16 e-13 0.06 e-13 3.55 e-13 0.80 e-13 1500 0.61 e-13 0.29 e-13 0.66 e-13 0.56 e-13 0.34 e-13

  13. Dense Rectangular matrices Dense Rectangular matrices

  14. Dense Rectangular matrices Dense Rectangular matrices

  15. Dense Rectangular matrices Dense Rectangular matrices • Error in singular values when m is fixed and n is varied n m Comb. Comb. Indirect Direct Francis Built in Square Method Method 0.42e ‐ 13 2.96e ‐ 13 2.42e ‐ 13 0.92e ‐ 13 2.59e ‐ 13 100 0.56e ‐ 13 0.37e ‐ 13 0.24e ‐ 13 0.56e ‐ 13 1.53e ‐ 13 200 700 700 0 92 0.92e ‐ 13 13 0 64 0.64e ‐ 13 13 0.71e ‐ 13 0 1 13 1.06e ‐ 13 1 06 13 2 49 2.49e ‐ 13 13 100 800 0.71e ‐ 13 0.60e ‐ 13 0.67e ‐ 13 1.56e ‐ 13 2.42e ‐ 13 0.99e ‐ 13 1.78e ‐ 13 1.20e ‐ 13 1.49e ‐ 13 2.98e ‐ 13 1400 0.71e ‐ 13 0.85e ‐ 13 0.63e ‐ 13 1.49e ‐ 13 5.76e ‐ 13 1500 100 0.67e ‐ 13 0.43e ‐ 13 0.32e ‐ 13 0.74e ‐ 13 2.63e ‐ 13 200 0.56e ‐ 13 87e ‐ 13 3.34e ‐ 13 1.24e ‐ 13 5.51e ‐ 13 0.78e ‐ 13 0.71e ‐ 13 0.71e ‐ 13 1.70e ‐ 13 8.6e ‐ 13 700 200 0.99e ‐ 13 0.78e ‐ 13 0.56e ‐ 13 1.45e ‐ 13 11.65e ‐ 13 800 1400 0.85e ‐ 13 0.92e ‐ 13 0.92e ‐ 13 2.06e ‐ 13 9.73e ‐ 13 1500 0.71e ‐ 13 1.28e ‐ 13 1.13e ‐ 13 1.92e ‐ 13 15.49e ‐ 13

  16. Dense Symmetric matrices Dense Symmetric matrices • Comparison of timings Comparison of timings Comb. Built in Comb. Square Indirect Direct Method Francis Built in SVD n Method 100 100 0.1478 0.1478 0.8 0.8 1.5 1.5 1.0154 1.0154 0.3 0.3 1.1 1.1 200 0.0396 3.3 6.9 4.3111 3 6.3 700 1.1737 359.7 414.4 57.9199 366.7 210.8 800 800 1.6642 66 2 603 603.4 6 675.6 6 77.6638 6638 622 2 622.2 3 3 313.4 1400 9.3720 5518.4 5761.6 73.7534 5655.7 1622.8 1500 11.2318 7256.0 7546.5 321.702 7405.2 1988.7

  17. Dense Symmetric matrices Dense Symmetric matrices • Comparison of Accuracies Comparison of Accuracies Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.35e ‐ 12 0 35e ‐ 12 0 62e ‐ 12 0.62e ‐ 12 1.171e ‐ 12 1 171e ‐ 12 0 258e ‐ 12 0.258e ‐ 12 0 91e ‐ 12 0.91e ‐ 12 200 0.83e ‐ 12 1.83e ‐ 12 8.027e ‐ 12 1.11e ‐ 12 2.59e ‐ 12 700 3.85e ‐ 12 12.88e ‐ 12 84.79e ‐ 12 24.28e ‐ 12 16.96e ‐ 12 800 4.66e ‐ 12 16.25e ‐ 12 75.42e ‐ 12 40.11e ‐ 12 20.10e ‐ 12 1400 10.53e ‐ 12 40.89e ‐ 12 781.6e ‐ 12 910.0e ‐ 12 51.26e ‐ 12 1500 10.92e ‐ 12 44.37e ‐ 12 133.4e ‐ 12 126.2e ‐ 12 55.05e ‐ 12

  18. Dense Symmetric matrices Dense Symmetric matrices • Comparison of Orthogonality checks ||UU T ‐ Id|| Comparison of Orthogonality checks ||UU ‐ Id|| Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.19e ‐ 13 0 19e ‐ 13 0 35e ‐ 13 0.35e ‐ 13 0.29e ‐ 13 0 29e ‐ 13 0 03e ‐ 13 0.03e ‐ 13 0 53e ‐ 13 0.53e ‐ 13 200 0.33e ‐ 13 0.69e ‐ 13 0.65e ‐ 13 0.06e ‐ 13 0.969e ‐ 13 700 0.89e ‐ 13 2.56e ‐ 13 2.52e ‐ 13 0.20e ‐ 13 3.40e ‐ 13 800 1.05e ‐ 13 2.87e ‐ 13 2.80e ‐ 13 0.22e ‐ 13 6.80e ‐ 13 1400 1.63e ‐ 13 5.94e ‐ 13 5.77e ‐ 13 0.36e ‐ 13 40.63e ‐ 13 1500 1.68e ‐ 13 7.04e ‐ 13 6.92e ‐ 13 0.38e ‐ 13 8.348e ‐ 13

  19. Dense Symmetric matrices Dense Symmetric matrices • Comparison of error in singular values Comparison of error in singular values Built in SVD Comb. Built in Comb. Square Indirect Method Direct Method n 100 100 0.03e ‐ 12 0 03e ‐ 12 0 421e ‐ 12 0.421e ‐ 12 0.29e ‐ 12 0 29e ‐ 12 0 12e ‐ 12 0.12e ‐ 12 0 13e ‐ 12 0.13e ‐ 12 200 0.07e ‐ 12 2.999e ‐ 12 0.65e ‐ 12 0.25e ‐ 12 0.78e ‐ 12 700 0.15e ‐ 12 27.92e ‐ 12 2.52e ‐ 12 0.49e ‐ 12 4.59e ‐ 12 800 0.14e ‐ 12 22.87e ‐ 12 2.80e ‐ 12 0.76e ‐ 12 8.27e ‐ 12 1400 0.18e ‐ 12 226.2e ‐ 12 5.77e ‐ 12 1.25e ‐ 12 12.32e ‐ 12 1500 0.22e ‐ 12 34.41e ‐ 12 6.92e ‐ 12 1.13e ‐ 12 21.03e ‐ 12

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