The root is double: ₊, only linear convergence.
Counterexample bank
Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.
AllODEPDEabeliananalyticityanovabairecalculus of variationscanonical formscauchy-riemanncompactnesscompletenesscomplexconformalconnectednesscontinuitycontour integrationconvergencedifferentiationdistributionsfieldsfixed pointgaloisgreengroupsharmonicinjectivityintegral equationsintegrationlaurentlimsuplinear algebraliouvillelpmarkovmeasuremechanicsmetric spacesmultivariablemultivariatenumericalp-groupspermutationspolynomialspower seriesprobabilityquadratic formsreal analysisregressionresiduesringssamplingseparabilityseparationsequencesseriessingularitiesstabilitystatisticssylowtestingtopologyuniform convergencewronskianzeros
#1 · ODE, PDE & Applied Mathematics › Numerical Analysis
“Newton's method converges quadratically to any root” — false
Counterexample: f(x) = x² at the root 0
numerical
#2 · ODE, PDE & Applied Mathematics › Numerical Analysis
“Newton's method always converges from any starting point” — false
Counterexample locked — unlock with Notes + PYQ
numerical
#3 · ODE, PDE & Applied Mathematics › Numerical Analysis
“Increasing the number of equally spaced interpolation nodes improves the approximation” — false
Counterexample: Runge's function 1/(1 + 25x²) on [−1, 1]
The interpolants diverge near the endpoints as . Chebyshev nodes restore convergence.
numerical
#4 · ODE, PDE & Applied Mathematics › Numerical Analysis
“A convergent method is stable for any step size” — false
Counterexample locked — unlock with Notes + PYQ
numericalstability