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 Tips and Techniques:
Numeric Solution of a Two-Point BVPRobert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Introduction</Font></Text-field>A recent query from a colleague in engineering posed a nonlinear two-point boundary value problem that could only be solved numerically. In this month's article, we'll explore the solution of a similar problem, highlighting the way Maple can be used to obtain the required solution.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Initializations</Font></Text-field>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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Statement of the BVP</Font></Text-field>Consider the nonlinear 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
and the related nonlinear boundary 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
that pose a two-point boundary value problem on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIzEuRidGL0Yv
<Text-field style="Heading 1" family="Arial" layout="Heading 1"><Font family="Arial">Using <Font italic="true">dsolve</Font> to Obtain a Solution</Font></Text-field>A naive use of Maple's dsolve command did not provide a solution to the problem originally posed, running at great length and then crashing. However, inclusion of the approxsoln parameter very quickly led to the desired solution. This parameter provides an initial guess for an iterative Rayleigh-Ritz solution process that converges very quickly to a solution. In the original engineering problem there were two possible solutions, only one of which was physically meaningful.If we apply that same strategy to the problem posed in the previous section, assuming that perhaps a solution might be close to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURjpGPQ==, we obtain the graph shown in Figure 1.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEob2RlcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEiUUYnRjZGOUZARitGKw==Figure 1 Solution of nonlinear two-point BVP starting with initial guess LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURjpGPQ==
In principle, it should be possible to solve this BVP via a shooting method, using the two boundary conditions as the governing equations that determine LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRImFGJ0YvRjJGPg== and 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. In the sequel, we apply this idea to the solution process, and also come to grips with another disturbing facet of the problem we selected, namely, finding all possible solutions.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">A Solution from First Principles</Font></Text-field>We begin by defining a procedure LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImJGJ0YvRjJGQUZBRkE= that, given values for 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, returns the list 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 This procedure implements a shooting method, starting from the initial values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, numerically integrating a solution for the resulting initial value problem, and returning the endpoint values.h := proc(a,b)
local F;
if type(a, numeric) or type(b, numeric) then
F := dsolve({q, y(0)=a,D(y)(0)=b}, y(x), numeric);
[op([2,2],F(1)),op([3,2],F(1))];
else 'h'(a,b);
end if;
end proc:
In terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY1Rjk=and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImJGJ0YvRjJGQUZBRkE=, the boundary conditions can be expressed via the 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
It is now possible to graph the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, resulting in Figure 2.
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 2 Graph of the equations for 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 and 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, as determined by the boundary conditions of the BVP
Figure 2 suggests that there are actually four solutions of the given BVP.Unfortunately, the fsolve command in Maple 11 does not permit systems of procedure calls, as would be required by our definitions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= through the procedure 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 Fortunately, this shortcoming is corrected in the next version of Maple. Meanwhile, we need to solve a pair of nonlinear algebraic equations whose terms are calculated via the solution of an initial value problem.We propose three methods for this. First, we implement Broyden's method as a generalization of the Secant algorithm for solving a single equation. Second, we apply a minimization technique to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlgtRiw2JS1GLzYlUSJnRidGMkY1RjhGQEY+, a sum of squares. Finally, because of the special nature of the first boundary condition, we can eliminate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= from the second equation, and apply the Secant method to the single resulting equation.In anticipation of all three iterative techniques, we define the following initial points, each taken from Figure 2.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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Broyden's Method</Font></Text-field>Newton's method for solving the system of equations 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 captured by the fixed-point 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNidRInhGJy8lJWJvbGRHRjEvRjBRJmZhbHNlRicvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkIvRjNRJ25vcm1hbEYnRkVGRQ== is the Jacobian matrix for F. Since this requires computation of derivatives, it would be tedious to implement a method for repeatedly computing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. As an alternative, we consider Broyden's method wherein LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is only computed once. Thereafter, an approximation to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is computed, an approximation that converges to 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRInhGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y6LUYjNiQtSSNtb0dGJDYtUSJeRicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGVUZDLyUnYWNjZW50R0Y0RkM= is a solution to the system 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After obtaining 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 either analytically or approximately by numeric differentiation, Broyden's method thereafter replaces 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as an approximation to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNidRInhGJy8lJWJvbGRHRjEvRjBRJmZhbHNlRicvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkUtRiM2JC1GLDYlUSJrRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRk1GTS1JI21vR0YkNi1RIi5GJ0ZNLyUmZmVuY2VHRkMvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGYG9GTQ==Our implementation of Broyden's method takes the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJnRidGNEY3Rj5GPi8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGPg== as impute, along with an initial point, a stepsize LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= used for numerically computing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, and the integer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= used to define the computational tolerance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21pR0YkNiVRIk5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==. Our code is simplistic, in that it does not type-checking, and does not make provision for generalization to other systems. It is provided as a sketch to show how such a computation might be implemented.Broyden := proc(F,X0,d,N)
local X, F0, F1, f, g, fa, ga, fb, gb, A0, A1, x0, x1, tol, k, L, Z;
x0 := X0;
X := <a,b>:
f := F[1];
g := F[2];
Z := Equate(X,x0);
F0 := eval(F,Equate(X,x0)):
fa := (eval(f,Equate(X,x0+<d,0>)) - F0[1])/d:
ga := (eval(g,Equate(X,x0+<d,0>)) - F0[2])/d:
fb := (eval(f,Equate(X,x0+<0,d>)) - F0[1])/d:
gb := (eval(g,Equate(X,x0+<0,d>)) - F0[2])/d:
A0 := Matrix([[fa,fb],[ga,gb]]):
x1 := x0 - A0^(-1).F0:
Digits := 30:
tol := 10^(-N):
for k from 1 to 10 do
F1 := eval(F, Equate(X,x1));
L := x1 - x0;
A1 := A0 + (F1 - F0 - A0.L).L^%T/(L.L);
Z := A1^(-1).F1;
if sqrt(Z.Z)<tol then break;
else x0 := x1; x1 := x1-Z; A0 := A1; F0 := F1;
end if;
end do:
Digits := 10:
x1;
end proc:
Using our Broyden procedure, we compute solutions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRImFGJ0YvRjJGPg== and 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.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
With pairs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPkY+ determined, we can solve the corresponding four initial value problems, and graph their solutions. The following commands generate the solutions and graphs,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
and the graphs themselves appear in Figure 3.
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 3 Graphs of four possible solutions generated as solutions of initial value problems from pairs 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 computed by Broyden's method
The four solutions found in Figure 3 can be obtained directly with Maple's dsolve command, provided an appropriate approximate solution is provided. If we list the four approximate 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
then the following loop computes the corresponding dsolve solutions.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
Figure 4 displays these solutions.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEob2RlcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjUTFGJ0Y2RjlGQEYrRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEob2RlcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjUTJGJ0Y2RjlGQEYrRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEob2RlcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjUTNGJ0Y2RjlGQEYrRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEob2RlcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjUTRGJ0Y2RjlGQEYrRis=Figure 4 Solutions obtained by dsolve using the approxsoln parameter
The graphs in Figures 3 and 4 are essentially the same.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Minimizing a Sum of Squares</Font></Text-field>A pair LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPkY+ that minimizes the sum of 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
is a solution to the equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJnRidGL0YyRjUtSSNtbkdGJDYkUSIwRidGOUY5. We can compute such minimizing pairs with the NLPSolve command in Maple's Optimization package. This command provides the nonlinear simplex (Nelder-Mead) method, which does not require computation of derivatives. The computed pairs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPkY+ and the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn at the minima are given by the following loop.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNipGKy1GIzYrLUkjbW9HRiQ2L1EkZm9yRicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNWVtRicvJSZkZXB0aEdGWi8lKmxpbmVicmVha0dRJWF1dG9GJy1GLDYlUSJrRicvJSdpdGFsaWNHRjsvRj1RJ2l0YWxpY0YnRlUtRjY2L1EjdG9GJ0Y5RjxGP0ZBRkRGRkZIRkpGTEZORlBGU0ZVLUkjbW5HRiQ2JFEiNEYnL0Y9USdub3JtYWxGJ0ZVLUY2Ni9RI2RvRidGOUY8Rj9GQUZERkZGSEZKRkxGTkZQRlNGKy1GVjYmRlhGZW5GaG4vRltvUTZpbmNyZWFzZWluZGVudG5ld2xpbmVGJy1GIzYnRistRiM2JS1GLDYlUSNTU0YnRmBvRmJvLUY2Ni1RI3xncnxnckYnRltwRkFGREZGRkhGSkZMRk4vRlFRLDAuMjIyMjIyMmVtRicvRlRGX3FGXW8tRjY2LVEjOj1GJ0ZbcEZBRkRGRkZIRkpGTEZOL0ZRUSwwLjI3Nzc3NzhlbUYnL0ZURmVxLUYjNiUtRiw2JVEpTkxQU29sdmVGJ0Zgb0Ziby1GNjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZbcEZBRkRGRkZIRkpGTEZORlBGUy1JKG1mZW5jZWRHRiQ2JC1GIzYsRmhwLUY2Ni1RIixGJ0ZbcEZBL0ZFRjtGRkZIRkpGTEZORlAvRlRRLDAuMzMzMzMzM2VtRictRiM2JS1GLDYlUS1pbml0aWFscG9pbnRGJ0Zgb0Ziby1GNjYtUSI9RidGW3BGQUZERkZGSEZKRkxGTkZkcUZmcS1GYHI2Ji1GIzYnRistRiM2JS1GLDYlUSJhRidGYG9GYm9GX3MtSSVtc3ViR0YkNiUtRiM2JS1GLDYlUSJQRidGYG9GYm9GW3FGXW8tRiM2Iy1GaG82JFEiMUYnRltwLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGZHItRiM2JS1GLDYlUSJiRidGYG9GYm9GX3MtRlx0NiVGXnQtRiM2Iy1GaG82JFEiMkYnRltwRmh0RitGW3AvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGZHItRiM2JS1GLDYlUSdtZXRob2RGJ0Zgb0Zib0Zfcy1GLDYlUTFub25saW5lYXJzaW1wbGV4RidGYG9GYm9GZHItRiM2JS1GLDYlUTBldmFsdWF0aW9ubGltaXRGJ0Zgb0Zib0Zfcy1GaG82JFElMTAwMEYnRltwRmRyLUYjNiYtRiw2JVE0b3B0aW1hbGl0eXRvbGVyYW5jZUYnRmBvRmJvRl9zLUYjNiUtRmhvNiRRJDEuMEYnRltwLUY2Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZbcEZBRkRGRkZIRkpGTEZORlBGUy1JJW1zdXBHRiQ2JS1GaG82JFEjMTBGJ0ZbcC1GaG82JFEkLTE1RidGW3AvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZqdEYrRitGW3BGK0YrLUZWNiZGWEZlbkZobi9GW29RNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnLUY2Ni9RJ2VuZH5kb0YnRjlGPEY/RkFGREZGRkhGSkZMRk5GUEZTRitGKw==
The first number returned in each solution is the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn at the computed extremum.To facilitate a comparison with the values computed by Broyden's method, we express the nonlinear simplex solutions as the vectorsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Oy1JI21vR0YkNi9RJGZvckYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRlYvJSpsaW5lYnJlYWtHUSVhdXRvRictRiw2JVEia0YnLyUnaXRhbGljR0Y3L0Y5USdpdGFsaWNGJ0ZRLUYyNi9RI3RvRidGNUY4RjtGPUZARkJGREZGRkhGSkZMRk9GUS1JI21uR0YkNiRRIjRGJy9GOVEnbm9ybWFsRidGUS1GMjYvUSNkb0YnRjVGOEY7Rj1GQEZCRkRGRkZIRkpGTEZPLUZSNiZGVEZXRlovRmduUTZpbmNyZWFzZWluZGVudG5ld2xpbmVGJy1GUjYmRlQvRlhRJjAuMGVtRidGWi9GZ25RKG5ld2xpbmVGJy1GUjYmRlRGYnBGWkZmbi1GLDYlUSJWRidGXG9GXm8tRjI2LVEnJnNwYXI7RidGZ29GPUZARkJGREZGRkhGSi9GTVEsMC4yNzc3Nzc4ZW1GJy9GUEZfcUZpbi1GMjYtUSomY29sb25lcTtGJ0Znb0Y9RkBGQkZERkZGSEZKRl5xRmBxLUYsNiVRJWV2YWxGJ0Zcb0Zeby1JKG1mZW5jZWRHRiQ2JC1GIzYlLUZocTYmLUYjNiYtRiw2JVEiYUYnRlxvRl5vLUYyNi1RIixGJ0Znb0Y9L0ZBRjdGQkZERkZGSEZKRkwvRlBRLDAuMzMzMzMzM2VtRictRiM2Iy1GLDYlUSJiRidGXG9GXm9GK0Znby8lJW9wZW5HUTMmTGVmdEFuZ2xlQnJhY2tldDtGJy8lJmNsb3NlR1E0JlJpZ2h0QW5nbGVCcmFja2V0O0YnRmNyLUklbXN1YkdGJDYlLUYjNiUtRiw2JVEjU1NGJ0Zcb0Zeby1GMjYtUSN8Z3J8Z3JGJ0Znb0Y9RkBGQkZERkZGSEZKL0ZNUSwwLjIyMjIyMjJlbUYnL0ZQRmB0RmluLUYjNiMtRmRvNiRRIjJGJ0Znby8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRmdvRistRlI2JkZURldGWi9GZ25RNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnRmBwRmZwLUYyNi9RJGVuZEYnRjVGOEY7Rj1GQEZCRkRGRkZIRkpGTEZPLUYyNi1RIn5GJ0Znb0Y9RkBGQkZERkZGSEZKRkxGT0Zpb0Yr
The Euclidean norms of the differences between the two solutions are given 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
The agreement is substantial.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Secant Method</Font></Text-field>If the first boundary condition is solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiYUYnRi9GMkY5RjlGOQ==, we 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
with which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can be eliminated from the second boundary condition. Thus, we have the single 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEjR2FGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIwRidGQEYrRis=
to solve. Figure 5 provides a graph of the left-hand side of this equation.
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Figure 5 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRImFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImJGJ0YvRjItRjY2JC1GIzYkRjpGQUZBRkFGQUZB
There are four intercepts for the graph in Figure 5, consistent with our earlier findings on the number of solutions of the nonlinear BVP. The equation 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 can be solved with the Secant method, which we have coded as the following procedure. Our procedure includes a calculation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, and returns both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==.Secant := proc(X0,N)
local y0,k,x0,x1,x2,y1,X,Y,tol;
Digits := 30:
tol := 10^(-N):
x0 := X0;
y0 := eval(Ga,a=x0):
x1 := 1.05*x0:
unassign('X','Y');
for k from 1 to 10 do
y1 := eval(Ga,a=x1);
x2 := x1 - y1 * (x1 - x0)/(y1 - y0);
if abs(x2-x1)<tol then X:=evalf[N](x2); Y:=evalf[N](eval(B,a=x2));break:
else x0:=x1; y0:=y1; x1:=x2; y1:=eval(B,a=x1);
end if;
end do:
Digits := 10:
<X,Y>;
end proc:
An application of the Secant method leads to the following four solutions.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
As we did earlier, we compare these solutions with those computed by Broyden's method.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 solutions appear to be identical.
Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.
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