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Piecewise quadratic bounding functions for finding real roots of polynomials

  • * Corresponding author: Djamel Aaid

    * Corresponding author: Djamel Aaid 
Abstract Full Text(HTML) Figure(3) / Table(4) Related Papers Cited by
  • In this paper, our main interest is to create/ construct a new useful and outstanding algorithm to obtain roots of the real polynomial represented by $ f(x) = c_{0}+c_{1}x+...+c_{i}x^{i}+...+c_{n}x^{n} $ where coefficients of the polynomials are real numbers and $ x $ is a real number in the closed interval of $ \mathbb{R} $. Also, our results are supported by numerical examples. Then, a new algorithm is compared with the others (writer classical methods) and this algorithm is more useful than others.

    Mathematics Subject Classification: 26C10, 90C20, 90C25, 90C90.

    Citation:

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  • Figure 1.  The underestimator in ($ BB $) and ($ BR $)

    Figure 2.  The underestimator in our method ($ BBR $)

    Figure 3.  Tightness of our underestimator $ p(x) $ than the $ q(x) $ used in the classical methods ($ BB $) and ($ BR $)

    Table 1.  Numerical results of Example 1

    Zeros of polynom Zeros found with BR Zeros found with BB Zeros found with BBR
    0.100000000 0.100000000 0.100000000 0.100000000
    0.200000000 0.199999999 0.199999999 0.199999999
    0.300000000 0.300000000 0.299999999 0.300000000
    0.400000000 0.400000000 0.400000000 0.400000000
    0.500000000 0.500000000 0.500000000 0.500000000
    0.600000000 0.600000000 0.600000000 0.600000000
    0.700000000 0.700000000 0.700000000 0.700000000
    0.800000000 0.800000000 0.800000000 0.800000000
    0.900000000 0.899999999 0.899999999 0.900000000
    Relative Error 0.000000002 0.000000003 0.000000001
    number of iterations 25 22 06
    Time (second) 0.082000000 0.010000000 0.000000000
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical results of Example 2

    Zeros of polynom Zeros found with BR Zeros found with BB Zeros found with BBR
    0.020600000 0.020599999 0.020600000 0.020600000
    0.056600000 0.056600000 0.056600000 0.056600000
    0.079900000 0.079900000 0.079899999 0.079899999
    0.210000000 0.210000000 0.209999999 0.210000000
    0.397300000 0.397300000 0.397299999 0.397300000
    0.446600000 0.446599999 0.446600000 0.446599999
    0.577600000 0.577600000 0.577599999 0.577600000
    0.955100000 0.955100000 0.955099999 0.955100000
    0.979100000 0.979100000 0.979099999 0.979100000
    0.983500000 0.983499999 0.983500000 0.983500000
    Relative Error 0.000000003 0.000000006 0.000000002
    number of iterations 27 32 12
    Time (second) 0.076100000 0.052000000 0.016000000
     | Show Table
    DownLoad: CSV

    Table 3.  Numerical results of Example 3

    Zeros of polynom Zeros found with BR Zeros found with BB Zeros found with BBR
    0.500000000 0.499999999 0.500000000 0.500000000
    0.250000000 0.249999999 0.249999999 0.250000000
    0.125000000 0.124999999 0.125000000 0.124999999
    0.062500000 0.062500000 0.062499999 0.062500000
    0.031250000 0.031249999 0.031250000 0.031249999
    0.015625000 0.015625000 0.015624999 0.015624999
    0.007812500 0.007812500 0.007812500 0.007812500
    0.003906250 0.003906250 0.003906250 0.003906250
    Relative Error 0.000000004 0.000000003 0.000000003
    number of iterations 36 27 11
    Time (second) 0.096200000 0.027300000 0.036100000
     | Show Table
    DownLoad: CSV

    Table 4.  Numerical results of Example 4

    Zeros of polynom Zeros found with BR Zeros found with BB Zeros found with BBR
    0.137793470 0.137793487 0.137793487 0.137793486
    0.729454549 0.729454566 0.729454566 0.729454565
    1.808342901 1.808342880 1.808342880 1.808342881
    3.401433697 3.401433705 3.401433705 3.401433705
    5.552496140 5.552496161 5.552496160 5.552496160
    8.330152746 8.330152734 8.330152735 8.330152735
    11.843785837 11.843785821 11.843785821 11.843785821
    16.279257831 16.279257837 16.279257837 16.279257837
    21.996585811 21.996585793 21.996585792 21.996585792
    29.920697012 29.920697000 29.920697001 29.920697002
    Relative Error 0.000000001 0.000000001 0.000000001
    number of iterations 23 13 07
    Time (second) 0.001200000 0.020000000 0.000000000
     | Show Table
    DownLoad: CSV
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