Do Numerical Methods Ass 2 Q1

Author: DoctorDalek1963

second-year/MA2K4-Numerical-Methods-and-Computing/Ass-2/main.tex

@@ -23,7 +23,48 @@
 Interpolate the function $f(x) = \sqrt x$ by a quadratic polynomial $p_2(x)$ with notes at $x_0 = 1$, $x_1 = 2$, and $x_2 = 3$. Isolate the coefficients in the polynomial. Further, compute the error at $x = 6$ to three significant digits.
 \end{questionbody}
 
-Answer
+\begin{align*}
+p_2(x) &= \sum_{k=0}^2 L_k(x) y_k \\[0.5ex]
+&= \sum_{k=0}^2 L_k(x) \sqrt k \\[0.5ex]
+&= L_0(x) \sqrt 1 + L_1(x) \sqrt 2 + L_2(x) \sqrt 3 \\[0.5ex]
+&= L_0(x) + L_1(x) \sqrt 2 + L_2(x) \sqrt 3 \\[0.5ex]
+&= \f{(x - x_1) (x - x_2)}{(x_0 - x_1) (x_0 - x_2)}
+    + \f{(x - x_0) (x - x_2)}{(x_1 - x_0) (x_1 - x_2)} \sqrt 2
+    + \f{(x - x_0) (x - x_1)}{(x_2 - x_0) (x_2 - x_1)} \sqrt 3 \\[0.5ex]
+&= \f{(x - 2) (x - 3)}{(1 - 2) (1 - 3)}
+    + \f{(x - 1) (x - 3)}{(2 - 1) (2 - 3)} \sqrt 2
+    + \f{(x - 1) (x - 2)}{(3 - 1) (3 - 2)} \sqrt 3 \\[0.5ex]
+&= \f{(x - 2) (x - 3)}{2}
+    + \f{(x - 1) (x - 3)}{-1} \sqrt 2
+    + \f{(x - 1) (x - 2)}{2} \sqrt 3 \\[0.5ex]
+&= \f{x^2 - 5x + 6}{2}
+    + \f{x^2 - 4x + 3}{-1} \sqrt 2
+    + \f{x^2 - 3x + 2}{2} \sqrt 3 \\[0.5ex]
+&= \f12 x^2 - \f52 x + 3
+    - \sqrt 2 x^2 + 4\sqrt 2 x - 3 \sqrt 2
+    + \f{\sqrt 3}2 x^2 - \f{3\sqrt 3}2 x + \sqrt 3 \\[0.5ex]
+&= \l( \f12 - \sqrt 2 + \f{\sqrt 3}2 \r) x^2
+    + \l( -\f52 + 4\sqrt 2 - \f{3\sqrt 3}2 \r) x
+    + \l( 3 - 3 \sqrt 2 + \sqrt 3 \r)
+\end{align*}
+
+The interpolation error at $x=6$ is
+\begin{align*}
+f(6) - p_2(6) &= \sqrt 6 - \l( \f12 - \sqrt 2 + \f{\sqrt 3}2 \r) 6^2 \\
+    &\qquad - \l( -\f52 + 4\sqrt 2 - \f{3\sqrt 3}2 \r) 6
+    - \l( 3 - 3 \sqrt 2 + \sqrt 3 \r) \\[0.5ex]
+&= \sqrt 6 - \l( 18 - 36 \sqrt 2 + 18 \sqrt 3 \r) \\
+    &\qquad + 15 - 24 \sqrt 2 + 9\sqrt 3
+    - 3 + 3 \sqrt 2 - \sqrt 3 \\[0.5ex]
+&= \sqrt 6 - 18 + 36 \sqrt 2 - 18 \sqrt 3 \\
+    &\qquad + 15 - 24 \sqrt 2 + 9\sqrt 3
+    - 3 + 3 \sqrt 2 - \sqrt 3 \\[0.5ex]
+&= -18 + 15 - 3
+    + 36 \sqrt 2 - 24 \sqrt 2 + 3 \sqrt 2 \\
+    &\qquad - 18 \sqrt 3 + 9\sqrt 3 - \sqrt 3 + \sqrt 6 \\[0.5ex]
+&= -6 + 15 \sqrt 2 - 10 \sqrt 3 + \sqrt 6
+\end{align*}
+This computes to be 0.342 to 3 significant digits.
 
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