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• W2189478049 abstract "This paper discusses the construction of new<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parameters<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>β</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>, and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>. The sufficient conditions for the positivity are derived on one parameter<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>while the other two parameters<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=normal>β</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>are free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation with<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivatives<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:mrow><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M12><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:math>. Comparisons with existing schemes also have been done in detail. From all presented numerical results the new<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M13><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated is<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M14><mml:mi>f</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>3</mml:mn></mml:mrow></mml:msup><mml:mfenced open=[ close=] separators=|><mml:mrow><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math>is also investigated in detail." @default.
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• W2189478049 title "Shape Preserving Interpolation Using<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle=italic>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>Rational Cubic Spline" @default.
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