Construction of Optimal Derivative-Free Techniques without Memory

<table><tr><td><table class="algorithm" id="alg3"><tr><td colspan="2"><span style="font-family:monospace">t</span><span style="font-family:monospace">ol = 8;</span></td></tr><tr><td colspan="2"><span style="font-family:monospace">initialValues = Sort</span><svg height="14.85" id="M365" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g>
</svg><span style="font-family:monospace">N</span><svg height="14.85" id="M366" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g>
</svg><span style="font-family:monospace">x /. </span>
            {
            <span style="font-family:monospace">ToRules</span><svg height="14.85" id="M367" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g>
</svg><span style="font-family:monospace">rts</span><svg height="15.05" id="M368" style="vertical-align:-2.30685pt" version="1.1" viewbox="0 0 11.8875 15.05" width="11.8875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.112)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g><g transform="matrix(.017,-0,0,-.017,5.927,12.112)"><path d="M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5
q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" id="x7D"></path></g>
</svg><span style="font-family:monospace">, tol</span><svg height="14.85" id="M369" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 11.85 14.85" width="11.85" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
</svg></td></tr><tr><td colspan="2"><span style="font-family:monospace">Length</span><svg height="14.85" id="M370" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g>
</svg><span style="font-family:monospace">initialValues</span><svg height="14.85" id="M371" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
</svg></td></tr><tr><td colspan="2"><span style="font-family:monospace">Accuracy</span><svg height="14.85" id="M372" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g>
</svg><span style="font-family:monospace">initialValues</span><svg height="14.85" id="M373" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 5.9875002 14.85" width="5.9875002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
</svg></td></tr></table></td></tr></table>

Journal of Applied Mathematics

alg3

Algorithm 3

Algorithm 3: Construction of Optimal Derivative-Free Techniques without Memory 