User:Egm6341.s11.team4.KTH/HW6

=Problem 6.4 Redo proof of HOTRE by trying to cancel terms with odd order of derivatives of g=

Given
From lecture note 30-2.

Let, $$ \ P_1(t) := -t $$

Then,

Objectives
1) Redo proof of Higher Order Trapezoidal Rule Error by trying to cancel terms with odd order of derivatives of g.

Solution
From the given equation.

Do integration by parts to eq(4.1)

To cancel terms with odd order of derivative of g, $$ \ P_2(1) = P_2(-1) = 0$$ in eq(4.2).

Do integration by parts again to eq(4.2).

Do integration by parts again to eq(4.4).

To cancel terms with odd order of derivative of g, $$ \ P_4(1) = P_4(-1) = 0$$ in eq(4.5).

Do integration by parts again to eq(4.6).

Do integration by parts again to eq(4.7).

Insert and combine eq(4.4), eq(4.5), eq(4.7) and eq(4.8) into eq(4.2).

After more integration by parts to eq(4.9).

Since $$ \ P_{2r+1}(1) = -P_{2r+1}(-1) $$ is odd function.

From the given equation.