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    Subsections


    Integration

    Abs-Exp

    Origin:Quateroni A, Sacco, R. and Saleri, F., Numerical mathematics, Springer 2000, pp. 413

    Compute

    \begin{displaymath}
\int_{-1}^{1}\vert x\vert e^{x} dx
\end{displaymath}

    The result is a range with width lower than $\epsilon$ that includes the value of the integral.

    Method: IntegrateRectangle or IntegrateTaylor
    Solutions::[1.2642411126774618, 1.2642411226375476] ( $\epsilon=1e-8)$
    Computation time (February 2006):

    DELL D400 (1.7Ghz) 0.019s (Rectangle) 0.004 (Taylor)

    comp.soft-sys.math.maple18257

    Origin: Article 18257 of comp.soft-sys.math.maple

    Compute

    \begin{displaymath}
\int_0^{2\pi} e^(\cos(x))\cos(x-\sin(x)) dx
\end{displaymath}

    Method: IntegrateTaylor
    Solution: [6.2831848095217104, 6.2831858068365776] ( $\epsilon=1e-6)$
    Solution:[6.2831853021819217, 6.2831853121792607] ( $\epsilon=1e-8)$
    Computation time (Mars 2006):
    DELL D400 (1.7Ghz), Taylor order: 3 0.22s ($\epsilon=1e-6$), 1.35s ($\epsilon=1e-8$)
    DELL D400 (1.7Ghz), Taylor order: 4 0.09s ($\epsilon=1e-6$), 0.21s ($\epsilon=1e-8$)
    DELL D400 (1.7Ghz), Taylor order: 5 0.09s ($\epsilon=1e-6$), 0.18s ($\epsilon=1e-8$)
    DELL D400 (1.7Ghz), Taylor order: 6 0.07s ($\epsilon=1e-6$), 0.18s ($\epsilon=1e-8$)

    Ellipsoid

    Origin:Quateroni A, Sacco, R. and Saleri, F., Numerical mathematics, Springer 2000, pp. 410-412

    Compute

    \begin{displaymath}
\int_0^{1/10}\sqrt{1-(100\,\sqrt {-2+2\,\sqrt {2}})x^2}dx
\end{displaymath}

    The result is a range with width lower than $\epsilon$ that includes the value of the integral.

    Method: IntegrateRectangle
    Solution: [.081356291802596431, .081357224879917173] ($\epsilon=1e-6$)
    Solution: [0.081356786414213011, 0.081356796338971332] ($\epsilon=1e-8$)
    Computation time (February 2006):

    DELL D400 (1.7Ghz) 0.001s ($\epsilon=1e-6$), 0.002s ($\epsilon=1e-8$)

    Sail boat

    Origin:Quateroni A, Sacco, R. and Saleri, F., Numerical mathematics, Springer 2000, pp. 410-412

    Compute

    \begin{displaymath}
\int_0^{10} \frac{\alpha x e^{-\gamma x}}{x+\beta} dx
\end{displaymath}

    with $\alpha =50$, $\beta =5/3$, $\gamma = 1/4$. The result is a range with width lower than $\epsilon$ that includes the value of the integral.

    Method: IntegrateRectangle
    Solution: [100.06136781918083, 100.06136881760561] ( $\epsilon=1e-6)$
    Solution:[100.06136831296580, 100.06136832296060] ( $\epsilon=1e-8)$
    Computation time (February 2006):

    DELL D400 (1.7Ghz) 0.0457s ($\epsilon=1e-6$), 0.886s ($\epsilon=1e-8$)

    sci.math.num-analysis89520

    Origin: Article 89520 of sci.math.num-analysis

    Compute

    \begin{displaymath}
\int_0^{\pi} \sqrt{1-cos^2(x)} dx
\end{displaymath}

    Method: IntegrateTaylor
    Solution: [1.99999527,2.0000004987] ( $\epsilon=1e-6)$
    Solution:[1.999999995087,2.0000000050096] ( $\epsilon=1e-8)$
    Computation time (Mars 2006):
    DELL D400 (1.7Ghz) 0.62s ($\epsilon=1e-6$), 1.33s ($\epsilon=1e-8$)

    sci.math.num-analysis.91892

    Origin: Article 91892 of sci.math.num-analysis
    with

     
    k = 400 * Pi / 3
    a = 3.75E-6
    q = 0.000875
    d1 = k*R * (3*a^2 - 2*R^2)
    d2 = 2*R^2 + a^2 * (k^2 * R^2 - 3)
    R = sqrt(a^2 + (q-z)^2)
    P = 0.005 / (0.17708 * 8*Pi^2*R^5
    
    Compute

    \begin{displaymath}
\int_{0.0025}^{0.0025} (P\cos(600\pi~)(d2\cos(kR)-d1\sin(kR))
\end{displaymath}

    Method: IntegrateTaylor
    Solution: [886.6380446722454,886.6380456712287] ( $\epsilon=1e-6)$
    Method: IntegrateTaylor at order $n$
    Computation time (August 2006):
    DELL D400 (1.7Ghz) 164.24s ($n=5$) 71.28s ($n=6$)

    stoutemyer2007

    Origin: [12]

    Compute

    \begin{displaymath}
\int_{0}^{1} log((exp(x)+2\sqrt{x}+1)/2.)
\end{displaymath}

    Method: IntegrateTaylor
    Solution: [0.67637670822541165607681348942, 0.67637670822543264594435536661], ( $\epsilon=2.1e-14)$
    Method: IntegrateTaylor at order $6$ for $x$ in $[10^{-8},1]$ and Integrate for $x$ in $[0,10^{-8}]$
    Computation time (November 2007):
    DELL D620 (1.7Ghz) 11s  


    next up previous Next: Other problems: inequalities, parametric Up: benches Previous: Optimization
  • J-P. Merlet home page
  • La page Présentation de HEPHAISTOS
  • HEPHAISTOS home page
  • La page "Présentation" de l'INRIA
  • INRIA home page

    jean-pierre merlet