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By default, 100 iterations are performed before FindRoot is aborted. The number of iterations performed before quitting is controlled by the option MaxIterations.
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MaxIterations n instructs Mathematica to use a maximum of n iterations in the iterative process before aborting.
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The equation e2x 2ex + 1 = 0 has x = 0 as its only root. However, because its multiplicity is 2, Newton s method converges very slowly. FindRoot[Exp[2 x] 2 Exp[x] + 1 0,{x, 100}]
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FindRoot cvmit : Failed to converge to the requested accuracy or precision within 100 iterations.
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50.} 4.54676 10 9}
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FindRoot[Exp[2 x] 2 Exp[x] + 1 0, {x, 100}, MaxIterations 300]
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FindRoot attempts to find real solutions. However, if a complex initial value is specified, or if the equation contains complex numbers, complex solutions will be sought. The equation in the next example has no real solutions.
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FindRoot[x2 + x + 1 0,{x, 2}]
FindRoot lstol: The line search decreased the step size to within tolerance speci ed by AccuracyGoal and PrecisionGoal but was unable to nd a suf cient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances.
0.500002}
FindRoot[x2 + x + 1 0,{x, I}]
0.5 + 0.866025 }
FindRoot[x2 + x + 1 0,{x, I}]
0.5 0.866025 }
There are three options that control the calculation in FindRoot and other numerical algorithms.
WorkingPrecision is an option that specifies how many digits of precision should be maintained internally in computation. The default is WorkingPrecision 16. AccuracyGoal is an option that specifies how many significant digits of accuracy are to be obtained. The default is AccuracyGoal Automatic, which is half the value of WorkingPrecision. AccuracyGoal effectively specifies the absolute error allowed in a numerical procedure. PrecisionGoal is an option that specifies how many effective digits of precision should be sought in the final result. The default is PrecisionGoal Automatic, which is half the value of WorkingPrecision. PrecisionGoal effectively specifies the relative error allowed in a numerical procedure.
EXAMPLE 19 We wish to obtain a 10-decimal place approximation to the solution of the equation cos 100 = x , x x +1 nearest to 5,000. FindRoot Cos 100 x , {x, 5000} x x +1
5000.83}
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Mathematica s defaults are insufficient to give the required accuracy. By increasing WorkingPrecision, we can obtain the desired result. FindRoot Cos 100 x , {x, 5000}, WorkingPrecision 28 x x +1
5000.83319115955609589817}
Since AccuracyGoal is, by default, half the value of WorkingPrecision, only the first 14 significant digits can be trusted. Thus, x 5000.8331911595 (accurate to ten decimal places).
EvaluationMonitor can be used to show intermediate calculations to be performed and displayed. The format is EvaluationMonitor expression.
The symbol can be found on the Basic Math Input palette or can be created by typing :>. This symbol is used instead of to avoid expression being immediately evaluated. This technique is illustrated in the next two examples.
EXAMPLE 20 To see how quickly the sequence of approximations converges when we solve the equation e x = x,
we can use EvaluationMonitor to print the results of intermediate calculations. n = -1; FindRoot[Exp[ x] x, {x, 2}, EvaluationMonitor {n++, Print[n," 0 1 2 3 4 5
", x]}]
2. 0.357609 0.558708 0.56713 0.567143 0.567143 0.567143}
EXAMPLE 21
To obtain a comparison between Newton s method and the secant method, we can ask EvaluationMonitor to print the number of iterations needed to converge to 100 significant digits. Newton s Method n = 0;
FindRoot[Exp[ x] x,{x, 1}, WorkingPrecision 100,
AccuracyGoal 100, EvaluationMonitor n++]
Print[n," iterations"]
0.5671432904097838729999686622103555497538157871865125081351310792230 457930866845666932194469617522946}
8 iterations Secant Method n = 0; FindRoot[Exp[ x] x,{x, 1, 2},WorkingPrecision 100, AccuracyGoal 100, EvaluationMonitor n++]
Print[n," iterations"]
0.5671432904097838729999686622103555497538157871865125081351310792230 457930866845666932194469617522946}
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