Numerical Methods: Equations Quiz

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Numerical methods for equations

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Suvalakshmi M

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Numerical Methods: Equations Quiz


Numerical Methods: Equations QuizVersión en línea Numerical methods for equations por Suvalakshmi M 1 Which condition guarantees a root exists in the Bisection Method? a f'(c) = 0 at some c in [a,b] b f(a) and f(b) have opposite signs c f(a) and f(b) are both zero d f(a) and f(b) are both positive 2 In the Regula Falsi (False Position) method, the next x value is found where the straight line through (a,f(a)) and (b,f(b)) crosses the axis. What is the formula for x_r? a x_r = b - f(b)(b-a)/(f(b)-f(a)) b x_r = a - f(a)/f'(a) c x_r = a - f(a)(b-a)/(f(b)-f(a)) d x_r = (a+b)/2 3 Newton–Raphson update formula is: x_{n+1} = ? a x_n + f(x_n)/f'(x_n) b x_n - f(x_n)/f'(x_n) c x_n - f'(x_n)/f(x_n) d x_n - f(x_n) 4 Gauss Elimination aims to transform the coefficient matrix to which form before back substitution? a Identity form b Row-echelon form c Diagonal form d Reduced row-echelon form only 5 How does Gauss–Seidel differ from Jacobi iteration for solving Ax=b? a Jacobi uses all updated values each iteration b Gauss-Seidel uses updated values as soon as they are available c Gauss-Seidel updates only the right-hand side d Both use only previous iteration values but Gauss-Seidel is faster by default 6 What is meant by the rate of convergence in iterative methods? a The accuracy of the derivative used b How fast the error decreases per iteration c The number of iterations required to reach machine precision d The size of the initial guess 7 Convergence of Gauss–Seidel is guaranteed for which type of matrices? a Symmetric positive definite (SPD) matrices b Any matrix c Non-singular but not SPD matrices d Diagonal matrices only 8 Which equation type is especially suited for Newton–Raphson when derivatives are easy to compute? a Linear equations only b Systems of equations only c Algebraic equations d Transcendental equations cannot use Newton–Raphson 9 What is a common stopping criterion for iterative root-finding methods? a \|f'(x_k)\| < tolerance b \|x_{k+1}-x_k\| < tolerance c \|f(x_k)\| > tolerance d Number of iterations exceeds a limit without check Feedback 1 A sign change in the interval implies a root by the Intermediate Value Theorem. 2 x_r is the x-intercept of the line joining (a,f(a)) and (b,f(b)). 3 Iterative root-finding using tangent line approximation. 4 Standard Gauss elimination leads to row-echelon form; Gauss-Jordan continues to reduced form. 5 Jacobi uses the previous iteration vector entirely; Gauss-Seidel uses newest components immediately. 6 Typical measures are order p and asymptotic error constant. 7 SPD matrices ensure convergence of Gauss–Seidel under standard norms. 8 Newton–Raphson leverages f'(x) readily for fast quadratic convergence. 9 Convergence is declared when successive iterates change by less than a tolerance.

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Numerical Methods: Equations QuizVersión en línea

por Suvalakshmi M

Which condition guarantees a root exists in the Bisection Method?

In the Regula Falsi (False Position) method, the next x value is found where the straight line through (a,f(a)) and (b,f(b)) crosses the axis. What is the formula for x_r?

Newton–Raphson update formula is: x_{n+1} = ?

Gauss Elimination aims to transform the coefficient matrix to which form before back substitution?

How does Gauss–Seidel differ from Jacobi iteration for solving Ax=b?

What is meant by the rate of convergence in iterative methods?

Convergence of Gauss–Seidel is guaranteed for which type of matrices?

Which equation type is especially suited for Newton–Raphson when derivatives are easy to compute?

What is a common stopping criterion for iterative root-finding methods?

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