Most commonly used emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, $mathcal{C}$, of non-spiky input functions in the Sobolev space $W^{2,infty}[a,b]$. Our algorithms automatically determine where to sample the function---sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function $f$ on a bounded interval with $L^{infty}$-error no greater than $varepsilon$ is $mathcal{O}Bigl(sqrt{{left|fright|}_{frac12}/varepsilon}Bigr)$ as $varepsilon to 0$. This is the same order as that of the best function approximation algorithm for functions in $mathcal{C}$. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if $f$ significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.
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