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Numerical methods formulae

The 4 results the numerical methods notes display, each with the conditions it needs to be true and a link to the lesson that states it. Of the 4, the formulae booklet prints 2 and 2 are working lines the notes derive in front of you.

Year 13 · 3 lessons with formulae.

A result the Edexcel formulae booklet prints is marked as such; one it does not print is marked to learn; and a result badged derived in the lesson is a working line the notes build in front of you rather than a standard result to memorise.

3 lessons

Locating roots and iteration

x_{n+1} = (x_{n} + 1)^{1/3}

NOT IN THE BOOKLET: DERIVED IN THE LESSONConditionsOne rearrangement of x³ − x − 1 = 0. Convergence needs |g′(x)| < 1 near the root, and another rearrangement of the same equation may not converge at all.

The Newton-Raphson method

x_{n+1} = x_{n} − \frac{f(x_{n})}{f'(x_{n})}

IN THE FORMULAE BOOKLETConditionsf differentiable, f′(xₙ) ≠ 0, and a starting value close enough to the root; a turning point nearby throws it off.

The trapezium rule

T = \frac{h}{2}(y_{0} + 2(y_{1} + … + y_{n−1}) + y_{n})

IN THE FORMULAE BOOKLETConditionsn strips of equal width h spanning [a, b].

L = h(y_{0} + … + y_{n−1}), \qquad R = h(y_{1} + … + y_{n})

NOT IN THE BOOKLET: DERIVED IN THE LESSONConditionsRectangle sums over n strips of equal width h, each strip roofed by one of its own end ordinates. They bound the integral only where the curve is monotonic across every strip: L is the lower bound and R the upper on an increasing stretch, and the two swap on a decreasing one. If the curve turns inside a strip, neither sum is a bound.

Every formula in the library is indexed by unit on the equation index, and the numerical methods notes are where these results are derived and used.

Also: Integration formulae · Vectors formulae