12-22-2019, 02:14 PM

An extract from In Defense of Linear Quadrature Rules, William Squire (Aerospace Engineering Dept., West Virginia University), Comp. & Maths with Apple, Vol. 7, pp. 147.-149, Pergamon Press Ltd., 1981

Abstract--lt is shown that appropiate linear quadrature rules can handle integrands with singularities at or near the end points more effectively than the nonlinear methods proposed by Werner and Wuytack. A special 10 point Gauss rule gives good results. A method with exponential convergence gives high accuracy with a moderate number of nodes. Both procedures were implemented on a programmable hand calculator.

INTRODUCTION

The purpose of this note is to demonstrate that:

(1) a special 10 point Gauss rule for integrands with singularities at or near the endpoints proposed by Harris and Evans [2] will give results comparing favourably to any other procedure using a comparable number of nodes.

(2) A quadrature rule, which Stenger [3] has shown to have exponential convergence, gives very accurate results for such integrands with a moderate number of function evaluations.

Both these procedures were implemented on an SR 52 programmable hand calculator.

…

The SR-52 implementation is given in Appendix A …

…

The method was implemented on an SR-52 as described in Appendix B …

…

APPENDIX A

Harris-Evans 10 point rule …

…

APPENDIX B

SR-52 program for Stenger quadrature (equation 1) …

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Abstract--lt is shown that appropiate linear quadrature rules can handle integrands with singularities at or near the end points more effectively than the nonlinear methods proposed by Werner and Wuytack. A special 10 point Gauss rule gives good results. A method with exponential convergence gives high accuracy with a moderate number of nodes. Both procedures were implemented on a programmable hand calculator.

INTRODUCTION

The purpose of this note is to demonstrate that:

(1) a special 10 point Gauss rule for integrands with singularities at or near the endpoints proposed by Harris and Evans [2] will give results comparing favourably to any other procedure using a comparable number of nodes.

(2) A quadrature rule, which Stenger [3] has shown to have exponential convergence, gives very accurate results for such integrands with a moderate number of function evaluations.

Both these procedures were implemented on an SR 52 programmable hand calculator.

…

The SR-52 implementation is given in Appendix A …

…

The method was implemented on an SR-52 as described in Appendix B …

…

APPENDIX A

Harris-Evans 10 point rule …

…

APPENDIX B

SR-52 program for Stenger quadrature (equation 1) …

BEST!

SlideRule