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Unit 2: Solution of Linear and Nonlinear Equations - Practice Quiz

ECE183 — Mathematics For Robotics 60 Questions
0 Correct 0 Wrong 60 Left
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1 Which equation defines an eigenvalue and a corresponding nonzero eigenvector of a matrix ?

Factorization based on eigenvalues Easy
A.
B.
C.
D.

2 If a square matrix is diagonalizable, which factorization can be used?

Factorization based on eigenvalues Easy
A.
B.
C.
D.

3 In the eigenvalue factorization , what does the diagonal matrix contain?

Factorization based on eigenvalues Easy
A. The eigenvectors of
B. The determinants of
C. The eigenvalues of
D. The singular values of

4 Which equation is commonly used to find the eigenvalues of a square matrix ?

Factorization based on eigenvalues Easy
A.
B.
C.
D.

5 Which expression represents the singular value decomposition of a matrix ?

Singular value decomposition Easy
A.
B.
C.
D.

6 In , where are the singular values of stored?

Singular value decomposition Easy
A. On the diagonal of
B. In every entry of
C. On the diagonal of
D. On the diagonal of

7 Which statement about the singular values of a real matrix is correct?

Singular value decomposition Easy
A. They are always equal
B. They are always complex
C. They are always negative
D. They are always nonnegative

8 The columns of in the decomposition are called what?

Singular value decomposition Easy
A. Characteristic vectors
B. Left singular vectors
C. Diagonal singular values
D. Right singular vectors

9 What is a root of a nonlinear equation ?

Solution of nonlinear equations Easy
A. A value where
B. A value where
C. A value where
D. A value where

10 Which of the following is a nonlinear equation?

Solution of nonlinear equations Easy
A.
B.
C.
D.

11 What condition is normally required to begin the bisection method on an interval ?

Bisection method Easy
A.
B.
C.
D.

12 How is the midpoint calculated in the bisection method?

Bisection method Easy
A.
B.
C.
D.

13 What happens to the interval containing the root after one bisection iteration?

Bisection method Easy
A. Its length is doubled
B. Its length becomes zero
C. Its length is halved
D. Its length is unchanged

14 How many initial approximations are required to start the secant method?

Secant method Easy
A. Four approximations
B. One approximation
C. Two approximations
D. Three approximations

15 What does the secant method use instead of the exact derivative ?

Secant method Easy
A. A midpoint value
B. A second derivative
C. A finite-difference slope
D. An integral value

16 Which formula gives the next approximation in the secant method?

Secant method Easy
A.
B.
C.
D.

17 Which formula is used in the Newton-Raphson method?

Newton-Raphson method Easy
A.
B.
C.
D.

18 What additional function information is required by the Newton-Raphson method?

Newton-Raphson method Easy
A. The first derivative
B. The maximum value of
C. The definite integral of
D. The inverse function

19 For , what is for use in the Newton-Raphson method?

Newton-Raphson method Easy
A.
B.
C.
D.

20 For and initial approximation , what is the next Newton-Raphson approximation ?

Newton-Raphson method Easy
A.
B.
C.
D.

21 For the matrix , which pair of matrices gives a valid factorization ?

Factorization based on eigenvalues Medium
A. ,
B. ,
C. ,
D. ,

22 A nonsingular matrix has eigenvalues and . What is ?

Factorization based on eigenvalues Medium
A.
B.
C.
D.

23 A diagonalizable matrix has eigenvalues and . What happens to as ?

Factorization based on eigenvalues Medium
A. approaches the zero matrix
B. alternates between two matrices
C. becomes unbounded in magnitude
D. approaches the identity matrix

24 What are the singular values of , listed in descending order?

Singular value decomposition Medium
A. and
B. and
C. and
D. and

25 A matrix has rank . In its compact SVD , what are the dimensions of , , and , respectively?

Singular value decomposition Medium
A. , , and
B. , , and
C. , , and
D. , , and

26 A matrix has singular values , , and . What is the Frobenius-norm error of its best rank-one approximation?

Singular value decomposition Medium
A.
B.
C.
D.

27 For and , what is the minimum-norm least-squares solution ?

Singular value decomposition Medium
A.
B.
C.
D.

28 For the fixed-point iteration , which local condition near a fixed point supports convergence?

Solution of nonlinear equations Medium
A. near
B. near
C. near
D. near

29 For a nonlinear system , Newton's method computes a correction . Which linear system should be solved?

Solution of nonlinear equations Medium
A.
B.
C.
D.

30 The equation is combined with . Which point in the first quadrant solves this nonlinear system?

Solution of nonlinear equations Medium
A.
B.
C.
D.

31 For , bisection begins on . After evaluating the first midpoint, which interval should be used next?

Bisection method Medium
A.
B.
C.
D.

32 An initial bisection interval has length . What is the minimum number of bisections needed to make the interval length at most ?

Bisection method Medium
A.
B.
C.
D.

33 Which condition guarantees that a continuous function has at least one root in the interval ?

Bisection method Medium
A.
B.
C.
D.

34 Starting with an interval of length , what is the interval length after exactly four bisection iterations?

Bisection method Medium
A.
B.
C.
D.

35 Apply one secant-method update to using and . What is ?

Secant method Medium
A.
B.
C.
D.

36 Why is the secant method useful when the analytical derivative of is difficult to obtain?

Secant method Medium
A. It approximates the derivative using two function values
B. It guarantees convergence from every initial pair
C. It requires only the sign of each function value
D. It replaces the function by a constant approximation

37 In the secant update, what is the main numerical concern when and are nearly equal?

Secant method Medium
A. The update denominator becomes very small
B. The two iterates become exact root bounds
C. The root automatically becomes a multiple root
D. The function becomes discontinuous at

38 Newton-Raphson is used to approximate by solving . If , what is ?

Newton-Raphson method Medium
A.
B.
C.
D.

39 For , why can Newton-Raphson not proceed directly from ?

Newton-Raphson method Medium
A. , causing division by zero
B. , making the function discontinuous
C. , so another root is required
D. , causing a repeated iterate

40 The function has a root of multiplicity . Which modified Newton update restores faster convergence when the multiplicity is known?

Newton-Raphson method Medium
A.
B.
C.
D.

41 Let . Which statement correctly describes the possibility of diagonalizing over ?

Factorization based on eigenvalues Hard
A. is diagonalizable because its only eigenvalue is positive
B. is not diagonalizable because its determinant is zero
C. is diagonalizable because every triangular matrix is diagonalizable
D. is not diagonalizable because its eigenspace has dimension

42 A real symmetric matrix has eigenvalues , , and . Which factorization and definiteness classification are correct?

Factorization based on eigenvalues Hard
A. and is positive definite
B. and is negative semidefinite
C. and is positive semidefinite
D. and is indefinite

43 Suppose is diagonalizable and has distinct eigenvalues , , and . What is the minimal polynomial of ?

Factorization based on eigenvalues Hard
A.
B.
C.
D.

44 Let . What is the best rank-one approximation to in the Frobenius norm?

Singular value decomposition Hard
A.
B.
C.
D.

45 For , which expression gives the Moore--Penrose solution of when ?

Singular value decomposition Hard
A.
B.
C.
D.

46 A matrix has singular values , , and . Which conclusion is valid for solving using an SVD-based pseudoinverse?

Singular value decomposition Hard
A. The condition number is infinite, and the matrix has no nonzero singular values
B. The nonzero-spectrum condition number is , and the null-space component is undetermined
C. The condition number is zero, and every solution is unique
D. The condition number is , and no least-squares solution exists

47 For the system , how many real solutions exist?

Solution of nonlinear equations Hard
A. No real solutions
B. Exactly one real solution
C. Exactly two real solutions
D. Infinitely many real solutions

48 For , which statement about the number of real roots is correct?

Solution of nonlinear equations Hard
A. It has exactly two real roots because its derivative has two zeros
B. It has exactly three real roots because its local extrema have opposite signs
C. It has exactly one real root because its degree is odd
D. It has no real roots because

49 Consider with domain excluding . A numerical method reports convergence to . What is the correct interpretation?

Solution of nonlinear equations Hard
A. is a valid double root because the numerator vanishes twice
B. The function has infinitely many roots near
C. The reported point is excluded from the domain and is not a root
D. is a valid simple root of the original equation

50 A continuous function satisfies on . After bisection iterations, what is the maximum possible interval width containing a root?

Bisection method Hard
A.
B.
C.
D.

51 For , a bisection method is initialized on . What is the midpoint after three completed iterations if the endpoint-value sign test is used?

Bisection method Hard
A.
B.
C.
D.

52 Which additional condition is required for bisection to guarantee convergence to a zero rather than merely relying on a sign change?

Bisection method Hard
A. The function must be continuous on the closed initial interval
B. The function must be strictly convex on the initial interval
C. The derivative must be positive throughout the interval
D. The function must be twice differentiable at every midpoint

53 A root is bracketed in an interval of width . What is the smallest number of bisection iterations needed to guarantee an absolute midpoint error below ?

Bisection method Hard
A.
B.
C.
D.

54 For the secant iteration applied to with and , what is ?

Secant method Hard
A.
B.
C.
D.

55 Assuming a simple root and suitable initial approximations, what is the asymptotic order of convergence of the secant method?

Secant method Hard
A. Quadratic, with order
B. Cubic, with order
C. Superlinear, with order approximately
D. Linear, with order

56 In a secant iteration, while . What occurs in the standard update?

Secant method Hard
A. The denominator vanishes, so the update is undefined
B. The next iterate is the midpoint of the two current points
C. The next iterate is exactly the known root
D. The method automatically becomes equivalent to Newton's method

57 Newton's method is applied to with . What is the local convergence behavior of the unmodified iteration?

Newton-Raphson method Hard
A. Cubic convergence to
B. Linear convergence with factor
C. Quadratic convergence to
D. Divergence for every initial point near

58 For , Newton's method started at produces which first two iterates?

Newton-Raphson method Hard
A. ,
B. ,
C. ,
D. ,

59 For , which initial value causes the Newton update to be undefined at the first step?

Newton-Raphson method Hard
A.
B.
C.
D.

60 For the nonlinear system , what is the Newton correction at ?

Newton-Raphson method Hard
A.
B.
C.
D.
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