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====== Math 753/853 HW2 ====== **Problem 1.** Write a Julia function ''bisectsearch'' that take a function $f(x)$ and an interval $a,b$ such that $f(a) f(b) < 0$, returns a root $r$ of $f$ within that interval, and makes a logarithmic plot of $|x_n -r|$ versus $n$. Find the root to as many digits accuracy as possible. //Challenges:// * Add checks to your ''bisectsearch'' function that verify that the starting conditions for bisection search are met. The checks should print a helpful error message and exit the function while returning the most reasonable value for the root from available information. * Think carefully about the stopping conditions for bisection search. What kinds of problems are possible for this algorithm when it's implemented in floating-point arthmetic? Add checks to your function that stop iteration and return the most reasonable value for the root. In some cases you might want to print an error or warning message. * Add some diagnostic printing to your function that prints out $a,b$, and $f(c)$ at each iteration. Add another argument ''diagnostics'' to the function that turns the printing on and off. Make ''diagnostics'' default to ''false''. * Write your ''bisectsearch'' function so that it operates equally well on any floating-point type: Float16, Float32, Float64, or BigFloat. * Instead of using ''if a; b; end'' statements for your checks, use the more Juliaesque ''a && b'' short-circuit conditional evaluation. Likewise, instead of using ''if a; b ; else c ; end'' statements, use the ''a ? b : c'' ternary operator. See [[http://docs.julialang.org/en/release-0.4/manual/control-flow/#man-conditional-evaluation]]. **Problem 2.** Apply the bisection search algorithm to find all real roots of **(a)** $3x^3 + x^2 = x + 5$ **(b)** $\cos^2 x + 6 = x$ **%%(c)%%** $\ln x + x^2 = 3$ more problems to come, plus Julia notebook...
