Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals

<table class="table-group" id="tab15"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td align="left" rowspan="2">Rules</td><td align="center" colspan="2">Integral</td></tr><tr class="thead"><td align="center">Approximate value of
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</svg></td><td align="center">Absolute error</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td align="left"><span style="width: 40.087502px;"><svg height="16.6625" id="M294" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 40.087502 16.6625" width="40.087502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center">1.46256</td><td align="center"><span style="width: 67.487503px;"><svg height="15.95" id="M295" style="vertical-align:-0.20474pt" version="1.1" viewbox="0 0 67.487503 15.95" width="67.487503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center">1.46268</td><td align="center"><span style="width: 67.487503px;"><svg height="15.9" id="M297" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 67.487503 15.9" width="67.487503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.017,-0,0,-.017,17.338,12.162)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,23.219,12.162)"><path d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
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</svg></span></td><td align="center">1.46266</td><td align="center"><span style="width: 67.487503px;"><svg height="15.9" id="M299" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 67.487503 15.9" width="67.487503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
<g transform="matrix(.012,-0,0,-.012,54.112,7.475)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,-0,0,-.012,61.097,7.475)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g>
</svg></span></td></tr><tr><td align="left" colspan="3"><hr/></td></tr><tr><td align="left">Exact value</td><td align="center">1.46265</td><td align="center">**</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

International Scholarly Research Notices

tab15

Table 15

Table 15: Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals 