today’s class boundary value problems eigenvalue problems numerical methods lecture 18 prof. jinbo...
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Today’s class
• Boundary Value Problems• Eigenvalue Problems
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Boundary Value Problems
• Auxiliary conditions• n-th order equation requires n conditions• Initial value conditions
• All n conditions are specified at the same value of the independent variable
• Boundary value conditions• The n conditions are specified at different values of
the independent variable
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Initial value conditions
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Boundary value conditions
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Types of Boundary Conditions• Simple B.C.: The value of the unknown function
is given at the boundary.
• Neumann B.C.: The derivative of the unknown function is given at the boundary.
• Mixing B.C.: The combination of the unknown function’s value and derivative is given at the boundary.
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Boundary Value Problems
• Shooting Method• Convert the boundary value problem into an
equivalent initial value problem by choosing values for all dependent variables at one boundary
Numerical MethodsLecture 18
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Shooting Method Example
• Convert to first-order
• Guess a value for z(0)• Use RK method or any other method to
solve the ODE at y=10
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Shooting Method Example• Using RK method with an initial guess of
z(0)=10 we get y(10)=168.3797• The specified condition was y(10)=200, so try
again• Using RK method with an initial guess of
z(0)=20 we get y(10)=285.8980• With two guesses and a linear ODE, we can
interpolate to find the correct value
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Shooting Method Example
z(0)=10 z(0)=20
z(0)=12.69
Numerical MethodsLecture 18
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Shooting Method
• If the ODE is non-linear, then you have to recast the problem as solving a series of root problems
• Think of the guessed values as a vector V
• That vector V will generate a y(yL) vector after solving the ODE
• The y(yL) vector should equal the boundary value conditions at yL
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Shooting Method
• Since y is a function of V, you can create a new discrepancy function F that is dependent on V
• The problem then becomes to zero the discrepancy
• We don’t know F so we cannot use normal root-solving methods
• But we can approach it by iteratively looking at the finite difference derivatives
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Finite-Difference Method (FDM)
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Finite-Difference Method (FDM)
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Finite-Difference Method (FDM)
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Matrix Form of Equation System
Numerical MethodsLecture 18
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FDM Example
Numerical MethodsLecture 18
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Eigenvalue Problems
• Eigenvalue problems are a special class of boundary-value problem
• Common in engineering systems with oscillating behavior• Springs• Vibrations
Numerical MethodsLecture 18
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Physics of Eigenvalues & Eigenvectors
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Physics of Eigenvalues
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Physics of Eigenvalues
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Steps of Solving Eigen-value Problems
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Eigenvalue Example
Numerical MethodsLecture 18
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Eigenvalue Example
Numerical MethodsLecture 18
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Boundary value eigenvalue problems• Polynomial Method
• Power Method
Numerical MethodsLecture 18
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Power Method
• The Power Method is an iterative procedure for determining the dominant (largest) eigenvalue of the matrix A and the corresponding eigenvector.
• Example
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Example of Power Method (Cont.)
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Example of Power Method (Cont.)
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Example of Power Method (Cont.)
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Example of Power Method (Cont.)
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Prof. Jinbo BiCSE, UConn
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Inverse Power Method
• The Power Method can be modified to provide the smallest (lowest) (magnitude) eigenvalue and its eigenvector. The modified method is known as the inverse power method
• The smallest eigenvalue λ in the matrix A corresponds to 1/ λ the largest eigenvalue in the inverse matrix A-1.
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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General I-P Method• The combination of Power Method and I-P
method can be use to determine all the eigenvalues and their related eigenvectors.
• Deflation Hotelling’s Method for symmetric matrices • First, normalize the largest eigenvector found
by the sum of the squares of the elements in the eigenvector
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General I-P Method
• Then create new A2 matrix
• The new A2 matrix has the same eigenvalues as A1 except that the largest eigenvalue has been replaced with 0
Numerical MethodsLecture 18
Prof. Jinbo BiCSE, UConn
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Hotelling’s Method
Numerical MethodsLecture 18
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Hotelling’s Method
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Next class
• Curve Fitting
Numerical MethodsLecture 18
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