/* MJ-9 Unit II: finite-difference identities for a cubic test function.
   Every printed residual must be zero. */
kill(all)$

f(x) := x^3-2*x+1$
forward : (f(x+h)-f(x))/h$
backward : (f(x)-f(x-h))/h$
centered : (f(x+h)-f(x-h))/(2*h)$
second_centered : (f(x+h)-2*f(x)+f(x-h))/h^2$

residual_forward : ratsimp(expand(forward-diff(f(x),x)-3*x*h-h^2))$
residual_backward : ratsimp(expand(backward-diff(f(x),x)+3*x*h-h^2))$
residual_centered : ratsimp(expand(centered-diff(f(x),x)-h^2))$
residual_second_centered : ratsimp(expand(second_centered-diff(f(x),x,2)))$
residual_example_first : ratsimp(subst([x=1,h=1/10],centered)-101/100)$
residual_example_second : ratsimp(subst([x=1,h=1/10],second_centered)-6)$

print("forward-difference residual =", residual_forward)$
print("backward-difference residual =", residual_backward)$
print("centered-first residual =", residual_centered)$
print("centered-second residual =", residual_second_centered)$
print("example first-derivative residual =", residual_example_first)$
print("example second-derivative residual =", residual_example_second)$