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.. _general_formulas:
General formulas and bounds
===============================================================================
This section collects some results from real and complex
analysis that are useful when deriving error bounds.
Beware of typos.
Error propagation
-------------------------------------------------------------------------------
We want to bound the error when `f(x+a)` is approximated by `f(x)`.
Specifically, the goal is to bound `f(x + a) - f(x)` in terms of `r`
for the set of values `a` with `|a| \le r`.
Most bounds will be monotone increasing with `|a|` (assuming that `x` is
fixed), so for brevity we simply express the bounds in terms of `|a|`.
**Theorem (generic first-order bound)**:
.. math::
|f(x+a) - f(x)| \le \min(2 C_0, C_1 |a|)
where
.. math::
C_0 = \sup_{|t| \le |a|} |f(x+t)|, \quad C_1 = \sup_{|t| \le |a|} |f'(x+t)|.
The statement is valid with either `a, t \in \mathbb{R}` or `a, t \in \mathbb{C}`.
**Theorem (product)**: For `x, y \in \mathbb{C}` and `a, b \in \mathbb{C}`,
.. math::
\left| (x+a)(y+b) - x y \right| \le |xb| + |ya| + |ab|.
**Theorem (quotient)**: For `x, y \in \mathbb{C}` and `a, b \in \mathbb{C}`
with `|b| < |y|`,
.. math::
\left| \frac{x}{y} - \frac{x+a}{y+b} \right|
\le \frac{|xb|+|ya|}{|y| (|y|-|b|)}.
**Theorem (square root)**: For `x, a \in \mathbb{R}` with `0 \le |a| \le x`,
.. math::
\left| \sqrt{x+a} - \sqrt{x} \right|
\le \sqrt{x} \left(1 - \sqrt{1-\frac{|a|}{x}}\right)
\le \frac{\sqrt{x}}{2} \left(\frac{|a|}{x} + \frac{|a|^2}{x^2}\right)
where the first inequality is an equality if `a \le 0`.
(When `x = a = 0`, the limiting value is 0.)
**Theorem (reciprocal square root)**: For `x, a \in \mathbb{R}` with `0 \le |a| < x`,
.. math::
\left| \frac{1}{\sqrt{x+a}} - \frac{1}{\sqrt{x}} \right|
\le \frac{|a|}{2 (x-|a|)^{3/2}}.
**Theorem (k-th root)**: For `k > 1` and `x, a \in \mathbb{R}` with `0 \le |a| \le x`,
.. math::
\left| (x+a)^{1/k} - x^{1/k} \right|
\le x^{1/k} \min\left(1, \frac{1}{k} \, \log\left(1+\frac{|a|}{x-|a|}\right)\right).
*Proof*: The error is largest when `a = -r` is negative, and
.. math::
x^{1/k} - (x-r)^{1/k} &= x^{1/k} [1 - (1-r/x)^{1/k}]
&= x^{1/k} [1 - \exp(\log(1-r/x)/k)] \le x^{1/k} \min(1, -\log(1-r/x)/k)
&= x^{1/k} \min(1, \log(1+r/(x-r))/k).
**Theorem (sine, cosine)**: For `x, a \in \mathbb{R}`, `|\sin(x+a) - \sin(x)| \le \min(2, |a|)`.
**Theorem (logarithm)**: For `x, a \in \mathbb{R}` with `0 \le |a| < x`,
.. math::
|\log(x+a) - \log(x)| \le \log\left(1 + \frac{|a|}{x-|a|}\right),
with equality if `a \le 0`.
**Theorem (exponential)**: For `x, a \in \mathbb{R}`,
`|e^{x+a} - e^x| = e^x (e^a-1) \le e^x (e^{|a|}-1)`, with equality if `a \ge 0`.
**Theorem (inverse tangent)**: For `x, a \in \mathbb{R}`,
.. math::
|\operatorname{atan}(x+a) - \operatorname{atan}(x)| \le \min(\pi, C_1 |a|).
where
.. math::
C_1 = \sup_{|t| \le |a|} \frac{1}{1 + (x+t)^2}.
If `|a| < |x|`, then `C_1 = (1 + (|x| - |a|)^2)^{-1}` gives a monotone bound.
An exact bound: if `|a| < |x|` or `|x(x+a)| < 1`, then
.. math::
|\operatorname{atan}(x+a) - \operatorname{atan}(x)| =
\operatorname{atan}\left(\frac{|a|}{1 + x(x+a)}\right).
In the last formula, a case distinction has to be made depending on the
signs of *x* and *a*.
Sums and series
-------------------------------------------------------------------------------
**Theorem (geometric bound)**: If `|c_k| \le C` and `|z| \le D < 1`, then
.. math::
\left| \sum_{k=N}^{\infty} c_k z^k \right| \le \frac{C D^N}{1 - D}.
**Theorem (integral bound)**: If `f(x)` is nonnegative and
monotone decreasing, then
.. math::
\int_N^{\infty} f(x) \le \sum_{k=N}^{\infty} f(k) \le f(N) + \int_N^{\infty} f(x) dx.
Complex analytic functions
-------------------------------------------------------------------------------
**Theorem (Cauchy's integral formula)**:
If `f(z) = \sum_{k=0}^{\infty} c_k z^k` is analytic (on an open
subset of `\mathbb{C}` containing the disk `D = \{ z : |z| \le R \}`
in its interior, where `R > 0`), then
.. math::
c_k = \frac{1}{2\pi i} \int_{|z|=R} \frac{f(z)}{z^{k+1}}\, dz.
**Corollary (derivative bound)**:
.. math::
|c_k| \le \frac{C}{R^k}, \quad C = \max_{|z|=R} |f(z)|.
**Corollary (Taylor series tail)**:
If `0 \le r < R` and `|z| \le r`, then
.. math::
\left|\sum_{k=N}^{\infty} c_k z^k\right| \le
\frac{C D^N}{1-D}, \quad D = \left|\frac{r}{R}\right|.
Euler-Maclaurin formula
-------------------------------------------------------------------------------
**Theorem (Euler-Maclaurin)**:
If `f(t)` is `2M`-times differentiable, then
.. math::
\sum_{k=L}^U f(k) = S + I + T + R
.. math::
S = \sum_{k=L}^{N-1} f(k), \quad I = \int_N^U f(t) dt,
.. math::
T = \frac{1}{2} \left( f(N) + f(U) \right) +
\sum_{k=1}^M \frac{B_{2k}}{(2k)!}
\left(f^{(2k-1)}(U) - f^{(2k-1)}(N)\right),
.. math::
R = -\int_N^U \frac{B_{2M}(t - \lfloor t \rfloor)}{(2M)!} f^{(2M)}(t) dt.
**Lemma (Bernoulli polynomials)**: `|B_n(t - \lfloor t \rfloor)| \le 4 n! / (2 \pi)^n`.
**Theorem (remainder bound)**:
.. math::
|R| \le \frac{4}{(2\pi)^{2M}} \int_N^U \left| f^{(2M)}(t) \right| dt.
**Theorem (parameter derivatives)**:
If `f(t) = f(t,x) = \sum_{k=0}^{\infty} a_k(t) x^k` and
`R = R(x) = \sum_{k=0}^{\infty} c_k x^k`
are analytic functions of `x`, then
.. math::
|c_k| \le \frac{4}{(2\pi)^{2M}} \int_N^U |a_k^{(2M)}(t)| dt.
|
