Nonlinear Equations

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Nonlinear Equations

• Multiple solutions
• Cant always be solved in closed-form
• Numerical solution techniques
• Newton-Raphson
• Continuation
• Optimization techniques

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Point-Slope Form of Line
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Solution Procedure

• Guess x0
• Evaluate f(x0)
• Check f(x0) 0 ?
• Evaluate df/dx at x0
• Evaluate x1 x0 Dx
• Evaluate f(x1)
• Check f(x1) 0 ?

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For n Equations, n Unknowns
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Two Equation Example
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Jacobian Matrix
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Linearized Equations
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For Initial Guesses
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Complex Linkages
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Complex Example

• Known Constants r, r2, r3, r5, x6
• Known Input Variable y6
• Unknown Variables Q2, Q3, x4, Q5

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Loop 1
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Loop 1 Equations
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Loop 2
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Loop 2 Equations
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Residual Functions
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Unknowns Jacobian
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