A) Halley's method B) Newton's method C) Householder's method D) Raphson's approximation
A) Joseph Raphson and Carl Friedrich Gauss B) Isaac Newton and Al-Khwarizmi C) Isaac Newton and Leonhard Euler D) Isaac Newton and Joseph Raphson
A) To integrate a complex function B) To calculate the derivative of a function C) To find a root of a function D) To find the maximum value of a function
A) Exact solutions without iteration B) Linear equations for curve fitting C) Larger derivatives of a function D) Better approximations to the roots of a function
A) An imaginary-valued function B) A purely constant function C) A step function only D) A real-valued function
A) An exact root and a tolerance level B) Its derivative and an initial guess C) Its second derivative and a final value D) Its integral and two boundary points
A) The y-intercept of the secant line at x0 B) The x-intercept of the tangent line at (x0, f(x0)) C) The inflection point of the curve D) The local minimum of the function near x0
A) The y-intercept of the linear approximation B) The slope of the linear approximation C) The maximum point of the linear approximation D) The unique root of the linear approximation
A) It increases linearly B) It triples with every step C) It increases by a constant factor of two D) It roughly doubles
A) Monte Carlo methods B) Householder's methods C) Euler's methods D) Runge-Kutta methods
A) Gradient descent B) Secant method C) Halley's method D) Bisection method
A) Only polynomial functions of degree two B) Only differential equations C) Complex functions and systems of equations D) Discrete probability distributions
A) The tangent line to the curve B) The secant line connecting the endpoints C) The vertical asymptote D) A horizontal line at f(xn)
A) f'(xn) must equal zero B) f'(xn) must equal one C) f'(xn) must be a complex number D) f'(xn) cannot equal zero
A) It diverges immediately B) It requires an infinite number of steps C) It results in a division by zero D) It generally requires fewer iterations to converge
A) Linear B) Cubic C) Logarithmic D) At least quadratic
A) Extra computations required for each step B) They cannot be used on computers C) They always diverge on real functions D) They require no derivatives at all
A) If the function has too many roots B) If the initial guess is too close to the root C) If the computer has too much memory D) If f or its derivatives are computationally expensive to evaluate
A) The ratio of f(xn) to f'(xn) B) The derivative divided by the function itself C) The sum of f(xn) and f'(xn) D) The product of f(xn) and f'(xn)
A) The exact final answer B) The absolute maximum C) The convergence threshold D) An initial guess
A) Renaissance period B) Old Babylonian period C) Victorian era D) Golden Age of Athens
A) 9th–11th century CE B) 1st–2nd century CE C) 16th–17th century CE D) 19th–16th century BCE
A) 17th–18th century CE B) 19th–16th century BCE C) 1st–2nd century CE D) 9th–10th century CE
A) Sharaf al-Dīn al-Ṭūsī B) François Viète C) Seki Kōwa D) Jamshīd al-Kāshī
A) 1427 B) 1680 C) 1633 D) 1711
A) 973–1048 B) 1540–1603 C) 1840–1900 D) 1135–1213
A) 1135–1213 B) 1633–1711 C) 1540–1603 D) 973–1048
A) John Wallis B) Arthur Cayley C) Henry Briggs D) Thomas Simpson
A) François Viète B) Seki Kōwa C) William Jones D) Joseph Raphson
A) 1821–1895 B) 1135–1213 C) 1648–1725 D) 1540–1603
A) Arthur Cayley B) Thomas Simpson C) John Colson D) Henry Briggs
A) 1687 B) 1711 C) 1671 D) 1669
A) Joseph Raphson B) William Jones C) John Colson D) John Wallis
A) 1671 B) 1685 C) 1669 D) 1690
A) John Colson B) William Jones C) John Wallis D) Thomas Simpson
A) Trigonometric equations B) Logarithmic equations C) Polynomials D) Differential equations
A) Error corrections B) Complex roots C) Derivative gradients D) Optimization factors
A) Constant terms B) The linear coefficients C) Higher-degree terms D) Initial root estimates
A) Geometric series B) Fourier series C) Binomial series D) Taylor series
A) Euler's equation B) Wave equation C) Heat equation D) Kepler's equation
A) Miftāḥ al-Ḥisāb B) De analysi per aequationes C) Philosophiæ Naturalis Principia Mathematica D) Method of Fluxions
A) 1633 B) 1740 C) 1711 D) 1687
A) Seki Kōwa B) Sharaf al-Dīn al-Ṭūsī C) François Viète D) Jamshīd al-Kāshī
A) The connection with calculus B) The use of error corrections C) The use of polynomials D) An initial estimate
A) Joseph Raphson B) Thomas Simpson C) John Wallis D) William Jones
A) Higher-degree terms B) The derivative function C) The original polynomial D) Taylor series expansion
A) A complex root formula B) A geometric optimization model C) A nonpolynomial Taylor series D) A reusable iterative expression
A) 1690 B) 1879 C) 1740 D) 1711
A) Extracting error corrections B) Finding roots of polynomials C) Solving optimization problems D) Approximating square areas
A) John Wallis B) Thomas Simpson C) Arthur Cayley D) Joseph Raphson
A) Faster B) Slower C) The same D) Quadratic
A) Decreasing the number of iterations B) Successive over-relaxation C) Removing the derivative D) Linear interpolation
A) The multiplicity B) The initial error C) The derivative D) The solution
A) m * x B) x / m C) xm / m D) m / x
A) Ten iterations B) Infinitely many iterations C) A single iteration D) Two iterations
A) No B) Yes C) Only for positive values D) Only for negative values
A) 1/4 x_n B) 1/2 x_n C) x_n2 D) 2 x_n
A) Exactly two B) Thousands C) Dozens D) Only one
A) Purely linear B) Quadratic C) Superlinear but subquadratic D) Cubic
A) About 2 B) About 4/3 C) About 1/2 D) About 10
A) Oscillating wildly B) Converging instantly C) Decreasing slowly D) Increasing rapidly
A) The 10th iterate B) The 50th iterate C) The 200th iterate D) The 2nd iterate
A) Convergent behavior B) Undefined behavior C) Linear behavior D) Oscillatory behavior
A) 0.06317 B) 0.99000 C) 1.00000 D) -1.76929
A) 1.00628 B) 1.00196 C) 0.03651 D) -0.06317
A) The derivative becomes zero everywhere B) Iterates can be sent outside the domain C) Oscillation stops permanently D) The method always converges faster
A) Global B) Universal C) Local D) Constant
A) Intermediate value theorem B) Mean value theorem C) Taylor's theorem D) Pythagorean theorem
A) α - xi B) xi + α C) xi - α D) xi / α
A) Negative B) Positive C) Zero D) Undefined
A) Undefined B) Negative C) Positive D) Zero
A) A vector B) A scalar C) A matrix D) A tensor
A) Trigonometric operations B) Basic arithmetic operations C) Matrix inversion D) Calculus integration
A) 305.5 B) 306.5 C) 612.5 D) 613.0
A) 5 decimal places B) 10 decimal places C) 2 decimal places D) 12 decimal places
A) 5mm B) 1mm C) 2mm D) 10mm
A) Minimum height B) Maximum thickness C) Minimum surface area D) Maximum volume
A) 1000000 B) 10000 C) 1000 D) 100000000
A) 2hbW = 1136000 B) hbw = 1136000 C) h + b + w = 1136000 D) hbw = 2000000
A) 1136000 / w B) 1136000 * hb C) 1136000 / (hb) D) 1136000 / (h + b)
A) [w, b] B) [b, w, h] C) [w, h] D) [b, h]
A) [1, 1] B) [0, 0] C) [50, 50] D) [100, 100]
A) [138.5684895, 65.67176536] B) [50.123, 110.456] C) [65.67176536, 138.5684895] D) [100, 100]
A) Jacobian matrix B) Vector matrix C) Identity matrix D) Hessian matrix
A) Integrals B) Asymptotes C) Derivatives D) Zeroes
A) Straight lines B) Circles C) Polygons D) Fractals
A) Curt McMullen B) Richard Hamilton C) Jürgen Moser D) John Nash
A) Hilbert space B) Euclidean space C) Fréchet space D) Banach space
A) Strictly positive B) Symmetric C) Boundedly invertible D) Continuously differentiable
A) 1960s B) 1950s C) 1940s D) 1970s
A) Complex initial conditions B) Smoothing operators C) Fréchet derivatives D) Universal starting points
A) 1980s B) 1970s C) 1960s D) 1950s
A) Complex analysis B) Fluid dynamics C) Celestial mechanics D) Quantum mechanics
A) John Nash B) Richard Hamilton C) Curt McMullen D) Jürgen Moser
A) Euclidean spaces B) Fréchet spaces C) Banach spaces D) Hilbert spaces
A) Second-order B) First-order C) Fourth-order D) Cubic
A) Maehly B) Pafnuty Chebyshev C) Newton D) Hensel
A) Maehly's procedure B) Hensel's lemma C) Quasi-Newton method D) Chebyshev's method
A) The entire field B) The unit ball C) The set of integers D) The set of primes
A) The Maehly analog B) The q-analog C) The p-adic analog D) The Chebyshev analog |