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# U of M MATH 5485 - Study guide for the final exam

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Study guide for the final examMath 5485, Fall 2008See sections 1-4 from study guide for the second midterm.5. Eigenvalues and eigenvectors, continued (Chapter 4)(a) Reduction to symmetric tridiagonal formi. Why first reduce to tridiagonal before calculating all eigenvalues and eigen-vectorsii. Properties of similarity transforms and orthogonal matrices,iii. How similarity transformation by a single Householder matrix put zeros incertain parts of matrixiv. How to combine these similarity transformations to turn into tridiagonalv. How eigenvectors are transformed in this process(b) Eigenvalues and eigenvectors of symmetric tridiagonal matricesi. The basic idea of how the QR algorithm worksii. Effect of pre- and post-multiplication by rotation matrix and how it underliesthe QR algorithmiii. How eigenvectors are transformed in this processiv. Wilkinson shift will not be on exam6. Interpolation (Chapter 5)(a) Basic ideasi. Interpolation versus approximationii. Why polynomials are good choice in principle (Weierstrass approximationtheorem)iii. With exception of solving tridiagonal system for cubic spline, be able to applythe interpolation scheme to word problems. Recognize if get bad results.(b) Lagrange form of interpolating polynomiali. Properties and formula of the Lagrange polynomials Ln,j(x)ii. Combining the Ln,jto interpolate function valuesiii. Uniqueness of interpolating polynomialiv. Interpolation errorv. Advantages and disadvantages of Lagrange form(c) Neville’s algorithmi. What Neville’s is good forii. Applying Neville’s algorithm to data(d) Newton’s form of interpolating polynomial1i. What Newton’s form is good forii. Divided differencesiii. Determining Newton’s form from data(e) Optimal points for interpolationi. Understand and use both definition and recurrence relation for Chebyshevpolynomialsii. Understand relationship between Chebyshev polynomials and smallest monicpolynomialsiii. Understand and be able to apply implications of analysis of Chebyshev poly-nomials on choosing optimal interpolating points for maximum normiv. Legendre polynomials and optimal interpolating points for Euclidean normwill not be on final exam(f) Piecewise polynomial interpolation (in general)i. Motivations for using different lower-order polynomials on each subintervalii. A partition(g) Piecewise linear interpolationi. Definition of piecewise linear interpolantii. Calculating piecewise linear interpolationiii. Error analysis(h) Cubic spline interpolationi. Definition of cubic spline and how it maximize smoothness of piecewise cubicii. Need for extra condition to solve for parametersiii. How one can calculate cubic spline efficiently (form tridiagonal system)iv. Differences among boundary conditions (not-a-knot, clamped, and natural)(i) Hermite interpolationi. Properties of polynomials Hi(x) andˆHi(x)ii. The Lagrange form of Hermite interpolating polynomialiii. The Newton form of Hermite interpolating polynomialiv. Calculating Hermite polynomial from data(j) Hermite cubic interpolationi. Properties of Hermite cubic interpolant (sacrificing smoothness for matchingderivative data)ii. Calculating Hermite cubic interpolant from

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