Question
A particularly efficient method for numerical integration of a function is the so-called Gaussian quadrature. In developing the formulas for this method it is necessary to find the roots of certain so-called Legendre polynomials. Numerically find all the roots of the sixth-order Legendre polynomial: P_6 (x)=1/48(693x^6-945x^4+315x^2-15)
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Answer to a math question A particularly efficient method for numerical integration of a function is the so-called Gaussian quadrature. In developing the formulas for this method it is necessary to find the roots of certain so-called Legendre polynomials. Numerically find all the roots of the sixth-order Legendre polynomial: P_6 (x)=1/48(693x^6-945x^4+315x^2-15)
Timmothy
4.888 Answers
1. Consider the polynomial given:
P_6(x) = \frac{1}{48}(693x^6 - 945x^4 + 315x^2 - 15)
2. To find the roots, set the polynomial equal to zero:
\frac{1}{48}(693x^6 - 945x^4 + 315x^2 - 15) = 0
3. Remove the fraction:
693x^6 - 945x^4 + 315x^2 - 15 = 0
4. Solve for \(x\):
The approximate numerical solutions are:
x \approx \pm 0.23861, \pm 0.66121, \pm 0.93247
2. To find the roots, set the polynomial equal to zero:
3. Remove the fraction:
4. Solve for \(x\):
The approximate numerical solutions are:
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