Numerical Methods 'Final' Questions
List three methods/ways that one can improve numerical estimation of a derivative.
Decrease step size Use higher order formula that employs more points Use Richardson Extrapolation
Explain two methods that can be employed effectively to solve a stiff initial value ordinary differential equation
Use implicit methods (Euler's methods) Use explicit methods with smaller step size taking care of stability criteria
If a function is a sixth order polynomial, how can one ensure that the numerical integration of the function gives the exact answer (explain the method that must be used)
Find I(2,1) and I(2,2) through Simpson's ⅓ and then find I(3,1) through adaptive quadrature
Explain how we can prove that a system of equations has no possible solution without graphing the functions
Find out the determinant of the systems post scaling. IF the determinant is zero no solution exist.
List two advantages of analytical methods as compared to numerical methods
Gives you an exact solution and no need for multiple iterations. Analytic solutions don't require long arithmetic.
Explain two advantages of Heun's method over Euler's method (6 points).
Heun's averages 2 slopes instead of taking just 1 value. Can also do multiple iterations before moving on
Explain how Heun's method is an improvement over Euler's Method
Heun's values uses averages of slope between first and following point. The Local Error is O(h^3) and Euler's is O(h^2)
Explain what method you would use to solve the following equation (1) to give error of true zero (true value) if the analytical answer is given by equation (2). (2nd order partial of y w.r.t. x) + 2*(1st order partial of y w.r.t x) = 12 (1) Y= x^4 + 6x^3 - 3x (2)
If boundary value problem then use finite difference method If initial value problem then use Runge Kutta with O(h^4) accuracy
Explain the two different techniques that can be used to solve a stiff ODE (6 points).
Implicit Euler - euler is simple equation/technique Implicit Hahn - more accurate than implicit euler because you can do more iterations before moving to the next i
Write a simple count loop in VBA format (2 points).
Solution For i=1 to 10 x= x+1 Next i OR Do while x<xmax x=x+1 Loop
List three characteristics that you would use to decide on the best numerical method for a given problem (3 points).
Solution a) Number of initial guess or starting point b) Rate of convergence c) Stability: will it always converge d) Accuracy and precision
Write an equation to express total error of a numerical solution if there is no analytical solution to the problem (2 points).
Solution: If not analytical solution and hence no true value, define approximate error (Ea)based on iterative process of solution: Ea = Current Approximation - Previous Approximation Approximate percent relative error: Ea=(current approx-previous approx)/current approx
List three main differences between analytical and numerical method (3 points)
Solution: a) Analytical methods are exact methods and the solution obtained is exact solution, while the numerical method is an approximate method with the solution obtained being close to the actual solution. b) Analytical method can be used for a limited class of problems while numerical methods can be used for almost all the problems. c) Significant amount of effort is wasted on the solution technique in analytical method. While in numerical method the emphasis is given to analysis and interpretation.
List three ways that error associated with Taylor series estimation of a function can be minimized
Solution: a) Reducing the step size b) Increasing the order of Taylor series (add more terms) . c) Make the equation more linear.
Show the condition that ensures the solution from Gauss-Seidel method converges. Explain where the condition was derived from
(a11) > (a12) and (a22) > (a21) ????? My guess is its derived from the determinant?
List two uses or applications of Taylor series (4 points).
Allows you to approximate any continuous polynomial and gives insight into truncation error
List two advantages of numerical methods as compared to analytical methods
Allows you to examine non-linear, large systems of equations and you can use sensitivity analysis
(a) What value of coefficient of determination (r2) indicates perfect fit of a polynomial regression equation to a set of data? (b) Explain why it is a perfect fit. (c) Explain when you would not use polynomial regression as compared to linear regression based on coefficient of determination (8 points).
A: r2 = 1, so Sr = 0 ..... r2 = (St - Sr) / (St) B: Sr = Sum (Yi measured - Yi modeled)^2 = Sum (Yi - Ao -A1Xi)^2 .... so Yi measured = Yi modeled C: If r2 = 1 for linear regression, then perfect fit....don't need to do any more
List three ways that total error associated with Taylor series estimation of a function can be reduced
A: reduce step size B: increase order of taylor series estimation (match order of function) C: avoid subtracting near like numbers D: work with smallest numbers first E: don't round off
Explain what method you would use to solve the following equation for the unknown, y given x of 3 (3 points). + 10 y=x2 y+ xe-x +10
As the equation is implicit in terms of y so any root finding method (bisection, false position, NewtonRaphson, Secant) can be used.
List two advantages that pivoting provide for solving linear system of equations
Avoids division by zero and minimizes round off error Also partially corrects an ill-conditioned system.
List and explain the error type that is only present in numerical solution of problem. Give one example of a method that can be used to determine that error (4 points).
Error associated with numerical solutions is : Truncation error: due to approximations in representing exact operations or quantities. Example is in Taylor series the truncation error is given by: Rn= fn+1(i)h^(n+1)/((n+1)!)
Explain the difference between local and global error. Explain the difference between the two will be negligible.
Local Error: results from application of the method over single step Global Error: it is submission of propagated truncation error and local error. When step size is very small
List and describe two practical and engineering reasons that an engineer would use numerical method to solve a non-linear system of equations to determine the best reservoir to produce if an analytical solution also exists (6 points).
Sensitivity Analysis: Sensitivity analysis is easier in case of numerical methods as b. Design: It is easier to determine the parameter when the required performance is compared to analytical method given with a numerical method.
Explain when Romberg integration or Richardson's Extrapolation can be used and how the error changes with use of this method compared to other numerical methods (6 points).
Must have equation or given data points with half step sizes. Error improves by power of 2 O(h2) to O(h4) or O(h4) to O(h6).
What are the principal two advantages of Newton-Raphson method compared to Secant method (4 points)?
Newton-Raphson method requires only one initial guess while the secant method requires 2 initial guesses. b) It converges faster as compared to secant method. c) Is more accurate as compared to secant method.
Given the following equation: f(x)=5e^x+(2/x)(e^x-1) Determine if numerical method or analytical method should be used to solve this problem. Explain your answer. If numerical method needs to be used, state the name of a method that can be used (6 points).
Numerical method should be used to solve this problem as it is an implicit equation in terms of x. Any root finding method (Newton-Raphson Method, Secant Method and Fixed Point Method) can be used.
Explain the main difference between regression and interpolation (4 points).
Regression a. Single curve that represents general trend designed to follow pattern, not intersect every point b. Applied to data exhibiting significant error Interpolation a. Fit a curve that pass directly through each of the points b. Applied to very precise data
Explain the difference between number of data points needed to derive a second-order polynomial equation based on regression versus interpolation. Also discuss the difference in terms of the accuracy of data points (4 points).
Regression: Need 3 or more data points. Can handle points with larger errors. Interpolation: Must be only three data points with very small errors.
1. Explain why Richardson Extrapolation method is more efficient than Trapezoidal rule. State when Richardson Extrapolation can be easily applied.
Richardson extrapolations is more efficient than Trap Rule because it gives more accurate integration solutions for very less number of functions to be solved.
Explain when Taylor series method can result in truncation error of zero for any kind of an equation (3 points).
Taylor series method can result in truncation error of zero for a nth order polynomial equation if nth order taylor series expansion is used. For other continues functions such as exponential, sinusoids etc. the truncation error can never be zero.
Describe two situations or conditions that would require use of numerical methods to determine the integral (4 points).
The equation cannot be integrated analytically. b. If only data points are given without the equation.
Describe two situations or conditions where finite element method is more efficient than finite difference method for solving PDE. Explain how the solution from finite element method is different from that of finite difference method.
The finite element method is used if - The system is having irregular geometry - Unusual boundary conditions - Heterogeneous compositions Finite difference method divides the solution domain into grid or discrete point or nodes. While the finite element divides the solution into simply shaped regions or elements. An approximate solution for the PDE can be developed for each of these elements and then assembling them together.
Describe the difference between local and global error when solving an ordinary differential equation. Which of these two errors is larger? Explain your reasoning
The local error reffers to the error incured over a single step. While the global error is the total discrepancy due to past as well as present steps. Anyone of them can be larger based on the sign of the error in each step.
Explain how optimum step size for a Taylor series estimation of a function was derived (i.e. what parameters does it depend on in words. Do not just write symbol) (4 points).
Total error is submission of round off and truncation error. To optimize the total error we will find the minimum of this submission
Why does each numerical derivative method result in reduction of total error between the estimate and true value as the step size is reduced, however the total error increases when the step size becomes too small (2 points)?
Total error is sum of truncation error and round off error. Usually round-off errors do not predominate so as the step size is reduced total error decrease due to reduction in truncation error Truncation error decreases while round-off error increases with smaller step size resulting in higher total error for too small step sizes.
Explain the fundamental difference between trapezoidal rule and simpson's ⅓ rule.
Trapezoidal is an integration method which considers a 1st order polynomial connecting the points. The Simpon's rule is the method that considers higher order polynomial (2nd ordered for simpons's ⅓ rule and 3rd order for simpson's 3/8th rule) connecting the points. Trapezoidal needs a single segment while simpson's method requires 2 equal segments for use.
List three different methods to solve a system of six linear equations. For each method, explain the condition of the system of equations that would make that method the most efficient
a) Gauss elimination: If the equation is not singular. b) LU elimination: When we do sensitivity analyses to see the effect of change in matrix B. c) Gauss-Seidel: reliable for diagonally dominant set of equations. d) Thomas Algorithm: Banded Matrix
List two shortcoming of the naive gauss elimination method and for each, explain what can be done to minimize or avoid the shortcoming
a. Division by zero- pivoting is done to minimize b. Roundoff error - scaling
Describe two different situations that would require using an open method rather than a bracketing method to solve a non-linear equation (6 points).
a. When multiple roots are present. b. When bracket for the roots is not available.
List steps that one would need to do to solve an ill-conditioned linear system of equations
ill conditioned ... d = 0 or close to zero solve: scale, pivot, solve by gaussian
