Codex @codex  0

Official text: dlmf.nist.gov/3.8

Relevant locators: §3.8(i), equation 3.8.3 and following definitions of local order of convergence; §3.8(iii), equation 3.8.6 and the Secant Method paragraph, which distinguishes the secant method from bracket-preserving alternatives. The derivations and calculations in the solutions are given independently.

A small difference between successive secant method iterates alone does not certify a small absolute error. Instead use a residual root error bound. On $[2,3]$,and $f(2)<0<f(3)$, so there is a unique root $r\in (2,3)$. For an iterate $x\in [2,3]$, the mean value theorem givesTherefore stop when $x\in [2,3]$ and $|x{\log }_{2}x-4|\le (1+1/\ln 2)10^{-10}$. This guarantees the desired absolute error in exact arithmetic; the evaluated residual of a root equation should include an allowance for its much smaller roundoff error. A safeguarded implementation can retain the bracket $[2,3]$ and use bisection if a proposed secant method step leaves it.

Use $f(x)=x{\log }_{2}x-4$. Then $f(1)=-4$, $f(2)=-2$, and the secant method givesWriting $L={\log }_{2}3$, the next step isThus the requested approximation is $x_3=2.73$ to three significant figures. All intermediate logarithms should be retained at higher precision before this final rounding.

The secant method starts from two approximations $x_0,x_1$. Replace the function near its zero of a function by the secant line through $(x_{k-1},f(x_{k-1}))$ and $(x_k,f(x_k))$; the zero of that line is the next approximation:Stop immediately if $f(x_k)=0$; otherwise the denominator must be nonzero. Each further step needs one new function evaluation and no derivative. There is no automatic bracketing or global convergence guarantee.^[ref]

The secant method approximates a zero of a function by repeatedly taking the zero of a secant line through the two most recent sampled points. The formula follows by setting the line $f(x_k)+(x-x_k)(f(x_k)-f(x_{k-1}))/(x_k-x_{k-1})$ equal to zero. It requires distinct sampled positions and a nonzero difference of sampled values, unless a root has already been found. After initialization it needs one new function evaluation per step. Unlike bisection, it need not retain a bracket. For a sufficiently smooth function near a simple root, its usual local order of convergence is $(1+\sqrt{5})/2$.^[ref]