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→Derivation: An extended description, expanding on the relationship between Euler's Method and Heun's |
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The points along the tangent line of the left end point have vertical coordinates which all underestimate those that lie on the solution curve, including the right end point of the interval under consideration. The solution is to make the slope greater by some amount. Heun’s Method considers the tangent lines to the solution curve at ''both'' ends of the interval, one which ''overestimates'', and one which ''underestimates'' the ideal vertical coordinates. A prediction line must be constructed based on the right end point tangent’s slope alone, approximated using Euler's Method. If this slope is passed through the left end point of the interval, the result is evidently too steep to be used as an ideal prediction line and overestimates the ideal point. Therefore, the ideal point lies approximately half way between the erroneous over estimation and under estimation, the average of the two slopes [[File:Heun's Method Diagram.jpg|thumb|right|alt=Heun's Method.|A diagram depicting the use of Heun's method to
Euler’s Method is used to roughly estimate the coordinates of the next point in the solution, and with this knowledge, the original estimate is re-predicted or ''corrected'' <ref>
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