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To what polynomial degree are the following rules exact?
Midpoint rule (� ¡ � )�
�+�
2
Trapezoidal rule
� ¡�
(� (�) + � (� ))
2
Simpson's
rule � ¡�
�+�
+ � (� )
� (�) + 4�
6
2
parabola
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Let �� ¡1 be an interpolant of � at nodes �1; :::; �� (of degree � ¡ 1)
Recall
Z �
X
�� ¡1(� )d� :
!�� (�� ) =
�
�
What can you say about the accuracy of the method?
����� Accuracy of Newton-Cotes
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Let �� be an interpolant of � at nodes �1; :::; �� (of degree � ¡ 1)
Recall
Z �
X
��(� )d�
!�� (�� ) =
�
�
What can you say about the stability of this method?
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What's not to like about Newton-Cotes quadrature?
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High-order polynomial interpolation requires a high degree of smoothness of the function.
����� Stitch together multiple lower-order quadrature rules to alleviate
smoothness requirement.
e.g. trapezoidal
What can we say about the error in this case?
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So far: nodes chosen from outside.
Can we gain something if we let the quadrature rule choose the weights,
too? ����� More design freedom ! Exact to higher degree.
����� Gaussian quadrature weight nder
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