CE2020 Final Exam: chapters 10 - 19
-Given a square matrix of n rows (and n columns) having elements ai,j, with i and j ranging from 1 to n, write down the formula for computing the Frobenius norm ||A||f.
((Sum[xi]^p)^1/p)
-Another measure of the spread of the data in a linear regression is that given by the coefficient of determination, 𝑟^2, and by the correlation coefficient, r. Provide the formula needed to compute the coefficient of determination, 𝑟^2, as a function of the variables 𝑆t and 𝑆r. Explain what the two variables involved in the formula for 𝑟 (i.e., 𝑆 and 𝑆) are.
(St-Sr)/St St is the standard deviation and Sr is the sum of squares
-You should know that performing interpolation with high-order polynomials can result in oscillations. Therefore, in the context of using polynomial interpolation...
, few terms (i.e., using low order polynomials) is better.
-In a least-square regression, 𝑆 is the sum of squares, which is computed as follows Sr = Sum(((yi -a0 -a1*xi)^2)) where 𝑎0 and 𝑎1 are best fit coefficients computed from what they are referred to as the normal equations. Write down the two (differential) equations that result from the minimization of 𝑆; also write down the resulting system of two linear algebraic equations resulting from these differential equations, which in turn allow to compute the two unknown coefficients 𝑎0 and 𝑎1.
-2 sum (yi-a0-a1xi) -2 sum ((yi-a0-a1xi)*xi)
-Consider an engineering model for which its behavior is represented by a system of linear algebraic equations of the form, [A]{x}={b}. Briefly explain each of the three methods that can be devised to determine if the model is ill conditioned.
1) Scale the matrix of coefficients [A] so that the largest element in each row is 1. Invert the scaled matrix and if there are several elements of [A]-1 that are several orders of magnitude greater then 1, it is ill conditioned. 2) Multiply the inverse by the original coefficient matrix, if the result is close to the identity matrix, it is not ill conditioned. 3) Invert the inverted matrix. If the result is not close to the original matrix, the system is ill conditioned.
Given a linear system of algebraic equations of the form [A]{x}={b}, what are the three steps involved in solving for the unknown vector {x} using the LU Factorization method? You are expected to produce a schematic representation of the process of solution using LU Factorization involving equations for the given matrix/vectors [A], {x} and {b}, and the new matrices/vector [L], [U] and {d}.
1. Elimination - [P][A] = [L][U] -Generate [U], generated by elimination with partial pivoting and storing the multiplier factors in [L] and employing the permutation matrix, [P] to keep track of the two switches. 2. Forward Substitution: The matrices, [L] and [P] are used to preform the elimination step with pivoting on {b} in order to generate the intermediate right hand side vector, {d}. This step can be represented concisely as the solution of the following matrix formula - [L]{d} = [P]{b} 3. Back Substitution: The final solution is generated in the same fashion as done before for Gauss elimination. this step can also be represented concisely as the solution of the matrix formulation - [U]{x} = {d}
-In a normal distribution (also referred to as a bell-shaped or Gauss distribution), what is the approximate percentage of measurements that is encompassed within a distance ±sy from 𝑦 (with sy being the standard deviation and 𝑦 being the arithmetic mean of the measurements)?
68%
-Write down the equation for the standard error of the estimate, sy/x, as a function of the sum of squares, 𝑆, and the number of data points, n. Briefly discuss what this standard error estimates.
= Sqrt (Sr/ (n-2)
-What does the norm of mathematical entities such as vectors and matrices indicate?
A norm is a real - valued function that provides a measure of the size or "length" of multicomponent mathematical entities such as vectors and matrices.
-What is a volume integral? Write down the mathematical equation for computing a volume integral of some problem relevant to engineering, explaining also what the chosen problem is about.
A volume integral is for summation and integration. equation is triple integral c (v) dV
-Where does the term quadrature comes from?
Archaic term that origionally meant the construction of a square having the same area as some curvilinear figure
-Write down the formula that defines the coefficient of variation, C.V. (in %), of a series of measurements in terms of the standard deviation sy and the arithmetic mean 𝑦.
C.V = sy/n x 100%
-What does the built-in MATLAB function chol() do? What are the arguments passed to the function? What does the function return?
Chol() will take the matrix that you set equal to and plus it into the equation [U]^T*[U] = X
-What is the formula to compute the matrix condition number, Cond[A], for example, in terms of the Frobenius norms for both matrix [A] and its inverse [A]-1? What does the matrix condition number measure?
Cond[A]= ||A|| * ||(A^-1)||
-Briefly explain the two methods that are available to perform non-linear regressions.
Gauss- Newton Method optimization methods
-Consider the mathematical equation 𝐼 = Integral(f(x),a,b) What does this equation stand for? i.e., how would you properly 'spell out' (or write 'with words') the equation?
I = integral from a to b f(x) dx. This stands for the integral of the function f(x) with respect to the independant variable x, evaluated between the limits x=a and x=b.
-With regard to the nature of engineering data being fitted, when should curve regression analysis be used? When should interpolation be used instead?
If the engineering data is a scatter plot then use a curve regression. Then if the plot has set points that are all along a curve then you are expected to use the interpolation method.
-What does the built-in MATLAB function lu() do? What are the arguments passed to the function? What does the function return?
If you use lu() function of a matrix. Then Matlab will give you the upper diagonal and lower diagonal matrix of the inputed matrix.
-In the context of fitting engineering data to a regression model, explain what non-linear models are. To illustrate your answer, give an example of a function (any function) of the type y = f (x) corresponding to a non-linear model.
In engineering normally fitting data is not going to be linear because your data is not all going to fall along one line. this is just not how the earth works. y = a0(1-e^-a1x)+e
-According to the dictionary definition, what does the term integrate mean?
Integrate- to bring together, as parts, into a whole; to unite; to indicate the total amount of.
-What is an areal integral? Write down the mathematical equation for computing an areal integral of some problem relevant to engineering, explaining also what the chosen problem is about.
Is a function of position. The equation is a double integral and the word flux dA
-If you have n measurements of engineering data, and these are denoted as yi, what does the following equation define? (1/n)Sumation(yi)
It determines the mean of a continuous function.
-Which is the most appropriate/efficient method of solution of linear algebraic systems of equations (of all those discussed in previous sections, namely, Graphical, Cramer's rule, Elimination of Unknowns, Naive Gauss Elimination, Pivoting Gauss Elimination, LU Factorization, Cholesky Factorization) that can be used to compute the inverse of a matrix? Why?
LU factorization. Provides an efficient means to evaluate multiple right hand side vectors.
-You should know that there are various ways for defining the equations to perform a polynomial interpolation.
Newton interpolation polynomials and Lagrange interpolation polynomials are two ways for defining these equations.
-With regard to solving a linear system of equations of the form [A]{x}={b} repeatedly, with {b} changing every time and with the matrix [A] remaining the same, why is the LU Factorization method preferred over the Gauss Elimination method?
Once Matrix A has been factored, in LU factorization it can be evaluated with multiple {b} values. While in Gauss we have decompose the Matrix every time!
-What is the difference between numerical integration and quadrature?
Quadrature is a synonym of numerical definite integration
-Write down the formula to compute the standard error, Sy/x, as a function of the sum of squares of the residuals, Sr, for n data points and a fitting polynomial of order m.
Sqrt[ Sr/ (n - (m+1))]
-Given a n-dimensional vector with elements x1, x2 ... xn, write down the formula for computing the Euclidean norm ||x||e.
Sqrt[Sum[xi^2]]
-Consider n data points corresponding to measurements of the dependent variable xi and the independent variable yi. The data points will be fitted using a second-order polynomial equation of the form: 𝑦 = 𝑎0+𝑎1* 𝑥+𝑎2*𝑥^2 + 𝑒 where e is the residual, and a0, a1 and a2 are the best fit coefficients to be determined. Write down the generic equation to compute the squares of the residual, Sr, in terms of the variables xi and yi for the n data points.
Sum ( yi - a0 - a1x - a2x^2)^2
Briefly discuss what the Gauss-Seidel method for solving a system of linear algebraic equations does. Illustrate your explanation for the case of a system of three equations (and three unknowns) of the form a11 x1 +a12 x2 +a13 x3 =b1 a21 x1 +a22 x2 +a23 x3 =b2 a31 x1 +a32 x2 +a33 x3 =b3 (your answer is expected to explain how these equations need to be re-written to implement the method, what initial guess values to use for the unknowns, what equation to use for checking the convergence of the solution, etc.).
The Gauss-Seidel method is the most commonly used iterative method for solving linear algebraic equation. Assume that we are given a set of n equations. [A]{x}={b} Suppose that for the conciseness we limit ourselves to a 3x3 set of equations. If the diagonal elements are all non zero, the first equation can be solved for x1, the second for x2 and the third for x3 to yield.
-Briefly explain the main difference between the Gauss-Seidel method and the Jacobi Iteration method for solving a system of linear algebraic equations.
The difference between the gauss - seidel method and Jacobi iteration is that for gauss you have to do step by set down through the iterations where as for jacobi you can solve from x1 with out the other two then plug into the final equations.
-If you have a continuous function f(x) that represents the behavior of an engineering model for the interval from a to b of the independent variable, what does the following mathematical expression define? (1/b-a)(Integral((f(x),a,b)dx)
Used to calculate the center of gravity of irregular objects and the root-mean-square current.
-Give an example of a physical or engineering problem for which interpolation of data may be required (you may want to read again Section 4.1 'Overview' of Part 4 in the textbook, where the fundamental difference between curve fitting and interpolation are discussed).
WE would use interpolation for waves. where we know the crest and the trough of each wave and therefore can interpolate the values inbetween.
-One way of solving a linear system of algebraic equations of the form [A]{x}={b} in MATLAB is to use the backslash (or 'left-division') operator as in >> x = A\b The other way is to use matrix inversion, with the built-in function inv(), as in >> x = inv(A)*b Explain how the backslash operator in the first way of solving the system works (e.g., in the internal MATLAB algorithm that implements the backslash operator, what is it checked first? what method is applied first in one case? what about the other case?).
When you use left division Matlab will first check is it is in a formate where the solution can be obtained without a full Gauss elimination. These include a. spaces /banded b. triangular c. symmetric. If any of these are detected the solution is obtained with that technique. However if there is no better solution then it will go through a full Gauss elimination with partial pivoting and the solution will be obtained with substitution.
-With regard to the matrix of coefficients [A], to what type of linear algebraic equations of the form [A]{x}={b} does the Cholesky Factorization method apply?
[A] = [U]^T*[U]
-If [A] is a square matrix, what is the inverse [A]-1 of this matrix (i.e., what is the fundamental relationship involvingthe original matrix and the inverse matrix)?
[A] [A]^-1 == [I]
-Write the matrix equation for a homogeneous and for a non-homogeneous system of linear equations.
[A]{x}=0 [A]{x}={b}
-In the LU Factorization method, what is the form of the matrix [U]? (e.g., is it a lower diagonal matrix? an upper diagonal matrix?) What is the form of the matrix [L]? (e.g., is it a lower diagonal matrix? an upper diagonal matrix? what are the values of the diagonal elements?).
[U] - upper diagonal matrix [L] - lower diagonal matrix The diagonal values are all 1
-You are expected to know how to write down the matrix equation that is characteristic of an eigenvalue problem (hint: this refers to equation 13.3 in the textbook). In this equation, you are expected to know that the parameter is and what the vector {x} is (for each value of ) is.
[[A]-(lambda)[I]]{x} = 0
-What does the histogram of a series of measurements provide? How is a histogram constructed? What does the horizontal and vertical axes in a histogram represent? What are the units of the horizontal and vertical axes in a histogram?
a bell curve of data is provided by a histogram. all of the data point are placed on the x axis and then the frequency is placed the y axis. data points - x frequency - y
-In the context of applying spline interpolation, what does the term knot refer to?
a knot is there for when the data points where two splines meet.
-Briefly discuss what the method of Successive Substitution for solving a system of non-linear equation does.
a method in which each of the nonlinear equations can be solved for one of the unknowns. These equations can then be implemented iteratively to compute new values which will converge on the solutions.
-In the context of having a set of data points from measurements (e.g., values of independent and dependent variables) from which an interpolation for a given value of the independent variable is required, explain what linear interpolation is.
a straight line fit between two given points.
-Consider an engineering model represented by a system of linear algebraic equations of the form [A]{x}={b}. If the matrix equation is regarded as a stimulus-response equation, the matrix equation can be written in the generic form as follows: [Interaction]{Response}={Stimuli}. Considering now a linear system of three equations and three unknowns, the elements of the vector {Response} can be computed as follows: x1 =a11-1 b1 +a12-1 b2 +a13-1 b3 x2 =a21-1 b1 +a22-1 b2 +a23-1 b3 x3 =a31-1 b1 +a32-1 b2 +a33-1 b3 where b1, b2 and b3 are the elements of the vector {Stimuli} and a11-1, a12-1, a13-1 are the elements of the inverse of the matrix [Interaction]. Based on the introduction above, discuss what the properties of the coefficients a11-1, a12-1, a13-1 in the equation above are, in view that an engineering model represented by a linear set of equations must satisfy the principles of superposition and proportionality.
a11-1 is a proportionality constant that gives the value of x1 due to a unit level of b1. A general conclusion can be drawn that aij-1 of the inverted matrix represents the value of xi due to a unit quantity of bj.
-In the context of polynomial interpolation, the general formula for a (n-1)th order polynomial is typically written as f(x) = a1+a2*x+a3*x^2 +...+an*(x^n-1) Write down the way in which the very same polynomial is represented in MATLAB (remember to denote the coefficients in the equation above as p, rather than a).
fminsearch[ function, x0, options, p1, p2 ...]
-What are first order (or linear) splines? To which type of polynomial interpolation within given intervals do the linear splines correspond?
have mearly a straight line connecting the two points.
-With regard to the resulting shape of the interpolated function, particularly at the knots, what is the disadvantage of using first-order splines?
higher order polynomials have smoother knots.
-How can the inverse of a square matrix be calculated? Illustrate your answer by discussing how different vectors of constants {b} need to be replaced in a system of linear equations of the form [A]{x}={b} to calculate the inverse [A]-1.
if you solve the matrix equation in with LU factorization that is below you will find the Inverse of A [A] {x} = [I]
-Briefly explain the main difference between interpolating engineering data using polynomial interpolation versus using (polynomial) spline functions; discuss also when interpolation using (polynomial) spline functions will work better than interpolation using polynomials.
in polynomial interpolation there is only one polynomial function used to st the entire set of data point in a spline function they use many different polynomial functions in a piece wise function to make the fit smoother and more exact.
-Briefly discuss the difference between extrapolation and interpolation.
interpolation is point to point while extrapolation starts at a point then grows predicting the future of the curve.
-What is the median (sometimes also called the 50th percentile) of a series of measurements?
is the mid point of a group of data
-You should know that polynomial interpolation...
is the most common method to interpolate (i.e., estimate intermediate values) between precise data points.
-What does the MATLAB function [n,xm]=hist(y,x) do? What are the variables n, xm, y and x in the expression [n,x]=hist(y,x)?
it creates a histogram of the data that you put into it. n- number of variables xm- a vector y - variable x - variable
-What is the origin of the term spline?
it was originally a drafting technique with a long thin strip that was used to draw smooth curves.
-You should be able to produce two diagrams showing some given data points (e.g., corresponding to some engineering measurements) and the corresponding best fit regression line, in one case, for large residual errors and in the other, for small residual errors.
line with close points line with far away points
-What are the mean and standard deviation values of a normally distributed series of random numbers generated with the command rndn()?
mean = 0 standard deviation = 1
-How many eigenvalues and how many eigenvectors will be for a square matrix of n×n?
n
-If m and s represent the mean and standard deviation values of a normally distributed series of random numbers to be generated with the MATLAB command randn(n), where n is the number of values to be generated, write down the commands to be typed in MATLAB to get the mentioned series of random numbers.
n + s * randn(n)
-You should know that when performing a (n-1)th polynomial interpolation...
n data points are required to find the n coefficients of the polynomial. For example, for a linear interpolation (polynomial of order 1), there are two unknown coefficients and two data points; for a quadratic interpolation (polynomial of order 2), there are three unknown coefficients and three data points; etc.
-Consider n data points to be interpolated with quadratic splines, for which three unknown constants exist for each interval. How many interpolation intervals do exist? How many number of unknowns do exist? Which are the four different conditions used to state the equations needed to solve for the total number of unknowns?
n-1
-What do the MATLAB functions norm() and cond() do? What are the arguments passed to the function?
norm(X,p) --> computes the norm cond(X,p) -->computes the condition number where X is the vector or Matrix and where p designates the type of norm or condition number.
-You are expected to be able to produce a sketch showing at least three cases for how integration is used to evaluate areas in engineering problems.
page 493
-In the context of polynomial interpolation, what do the MATLAB functions polyfit() and polyval() do? What are the arguments passed to the function? What are the values returned by the function?
polyfit() create the best polynomial for the given data set. polyval() evaluates the polynomial at each point x.
-What does the MATLAB function polyfit() do? What are the arguments passed to the function? In which order do the coefficients of the polynomial need to be passed to the function? What does the function return?
polyfit() will fit a least squares regression to the nth order polynomial to the data. you need to pull in and x and y vector and then the order of the polynomial n.
-What is the difference between the MATLAB functions rand() and randn()? How would the histograms of a series of random values generated by the functions rand() and randn() be? i.e., what type of distribution function will apply to each case?
rand is uniformly distributed distributed between 0 and 1. randn with a mean of 0 and a standard deviation of 1
-You are expected to know how to compute the two eigenvalues for a square matrix of 2×2 using the polynomial method, as outlined in equations (13.5) through (13.7) in the textbook.
take the determinant of the 2 by 2 matrix and solve as if it was equal to zero.
-What is the arithmetic mean of a series of engineering measurements? If the number of measurements is n, and each measurement is denoted as yi, what is the formula needed to compute the arithmetic mean 𝑦 ?
the arithmetic mean of a sample s defined as the sum of the individual data points divided by the number of points or where the summation is from i=1 through n. formula: y(arth) = Sum (yi)/n
-When n measurements have been made and the standard deviation is being computed, why is the denominator in the formula for the standard deviation sy equal to n-1 and not equal to n?
the degrees of freedom is n-1 and that is hat you are dividing by.
-Briefly explain what a diagonally dominant system of linear equations is.
the diagonal coefficients in each of the equations must be larger than the sum of the app solute values of the other coefficients in the equation.
-What does the built-in MATLAB function fsolve() do? What are the arguments passed to the function? What are the results returned by the function?
the fsolve[] function solves the system of nonlinear equations with several variables. A general representation of its syntax... [x,fx] = fsolve(function, x0, options)
-What is the mode of a series of measurements?
the middle number
-What is the standard deviation about the mean for a series of measurements? If n is the number of measurements, yi refers to each measurement, and 𝑦 refers to the arithmetic mean of the measurements, what is the formula needed to compute the standard deviation sy?
the most common measure of the spread for a sample is the standard deviation about the mean sy = [Sum(yi)^2 - (Sum(yi^) /n)]/n-1
With regard to the coefficient of determination, 𝑟^2 (and the the correlation coefficient, r) briefly discuss what the lower and upper values of 𝑟^2 (and r) are; also discuss what the resulting value of 𝑟^2 (or r) tells about the quality of the best fit line.
the numbers range between 0 and 1 where 0 is no correlation and 1 is a perfect correlation
-What are quadratic (or second-order) splines?
they are higher order splines used to contract smooth and continuous knots.
-What is the relationship between first order spline interpolation and (what is it known as) table lookup?
they are the same
-What do the MATLAB functions mean(s), median(s), mode(s), min(s) and max(s), with s being a vector containing engineering data, do?
they give you what ever you are asking for of the set of s.
-What does the MATLAB function rand(m,n) do? What are the variables m and n in the expression rand(m,n)? What are the maximum and minimum values returned by this function?
this computes a random matrix in with m and n are the size of the matrix like mXn
-You should know that a quadratic or parabolic (i.e., second-order polynomial) interpolation can be performed when...
three data points are given and that this case of interpolation corresponds to a quadratic (or parabolic) function passing through the three points.
-You should know that a linear (i.e., first-order polynomial) interpolation can be performed when...
two data points are given and that this case corresponds to considering a straight line that passes through the two points.
-Given a saturation-growth-rate-equation model of the type 𝑦 𝑎𝑥/𝑏 𝑥, explain how this model can be linearized so that a simpler linear least-square method can be applied in the (regression) analysis. Your explanation should address how the saturation-growth-rate-equation model should be recasted (i.e., what type of transformation is needed in both horizontal and vertical axes when representing the model) and how the slope and intersect of the best fit line will be related to coefficients 𝛼 and 𝛽 in the saturation-growth-rate-equation model. You are also expected to produce a schematic diagram of the saturation-growth-rate-equation model to illustrate your answer.
y = a *((x)/(B+x)) you can break it up using the law of logs which states that if you are dividing a set of equations then you can place a log on both sides and take the division and turn it into subtraction.
-Given a power-equation (regression) model of the type type 𝑦 𝛼 𝑥 , explain how this model can be linearized so that a simpler linear least-square method can be applied in the (regression) analysis. Your explanation should address how the power-equation model should be recasted (i.e., what type of transformation is needed in both horizontal and vertical axes when representing the model) and how the slope and intersect of the best fit line will be related to the coefficients 𝛼 and 𝛽 in the power-equation model. You are also expected to produce a schematic diagram of the power-equation model to illustrate your answer.
y = ax^B if you place a natural log on both sides that will end up with a linear model
a simpler linear least-square method can be applied in the (regression) analysis. You explanation should address how the exponential model should be recasted (i.e., what type of transformation is needed in both horizontal and vertical axes when representing the model) and how the slope and intersect of the best fit line will be related to the coeffiecients 𝛼 and 𝛽 in the exponential model. You are also expected to produce a schematic diagram of the exponential model to illustrate your answer.
y= a1e^B1x
-In the polynomial method, which is the matrix equation that allows to solve for the different eigenvalues ? (hint: this refers to equation 13.4 in the textbook)
| [A] - (lambda)[I] | = 0
-With reference to diagonally dominant system of linear equations, when will the Gauss-Seidel method for solving the system converge? When will it not converge?
||(aii)|| > Sum|aij| convergence is guaranteed if the condition is satisfied. if this condition is not met then it will diverge.
