Given the polynomial P (x) = x4 + 5x3 − 9x2 − 85x − 136
Use Newton’s method with Horner to find a root with ǫ = 10−5, starting from x0 = −4
If xR is the solution found before, find the polynomial P1(x) obtained by dividing the original polynomial by x − xR .
Again use Newton’s method with Horner to find a root of P1(x).
Verify that the root found is also a root of P (x).
Use Newton’s Method to find a solution of the equation e6X + 3(ln 2)2e2X − e4X ln 8 − (ln 2)3 = 0 with error tolerance 10−5, and that is in the interval
−1 ≤ x ≤ 0.
Repeat the previous exercise using the Secant Method.
For each one of the following systems of linear equations:
I)
= 8x1 + 3x2
= 12x2 + 6x3
= x1 + 10x3
II)
2x1 + x2 + 5x3 = 1
2x1 + 2x2 + 2x3 = 1
4x1 + x2 = 2
III)
x1 + x2 − x3 = −3
6x1 + 2x2 + 2x3 = 2
−3x1 + 4x2 + x3 = 1
Use Gaussian Elimination (2.0) with backward substitution to find the solution. Show the resulting matrices after each one of the matrix row operations.
Use Cramer’s Rule to solve them. Compute the determinants using minors.
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