(geometric-estimate-seq S-fn next-S-fn factor n0)Accepts:
S-fn: a function of n that generates a numerical integral estimate from
n slices of some region, andnext-S-fn: a function of (previous estimate, previous n) => new estimatefactor: the factor by which n increases for successive estimatesn0: the initial n to pass to S-fnThe new estimate returned b next-S-fn should be of factor * n slices.
Accepts: - `S-fn`: a function of `n` that generates a numerical integral estimate from `n` slices of some region, and - `next-S-fn`: a function of (previous estimate, previous `n`) => new estimate - `factor`: the factor by which `n` increases for successive estimates - `n0`: the initial `n` to pass to `S-fn` The new estimate returned b `next-S-fn` should be of `factor * n` slices.
(incrementalize S-fn next-S-fn factor n)Function that generalizes the ability to create successively-refined estimates of an integral, given:
S-fn: a function of n that generates a numerical integral estimate from
n slices of some region, andnext-S-fn: a function of (previous estimate, previous n) => new estimatefactor: the factor by which next-S-fn increases n in its returned estimaten: EITHER a number, or a monotonically increasing sequence of n slices to use.If n is a sequence, returns a (lazy) sequence of estimates generated for
each entry in n.
If n is a number, returns a lazy sequence of estimates generated for each
entry in a geometrically increasing series of inputs $n, n(factor),
n(factor^2), ....$
Internally decides whether or not to use S-fn or next-S-fn to generate
successive estimates.
Function that generalizes the ability to create successively-refined estimates of an integral, given: - `S-fn`: a function of `n` that generates a numerical integral estimate from `n` slices of some region, and - `next-S-fn`: a function of (previous estimate, previous `n`) => new estimate - `factor`: the factor by which `next-S-fn` increases `n` in its returned estimate - `n`: EITHER a number, or a monotonically increasing sequence of `n` slices to use. If `n` is a sequence, returns a (lazy) sequence of estimates generated for each entry in `n`. If `n` is a number, returns a lazy sequence of estimates generated for each entry in a geometrically increasing series of inputs $n, n(factor), n(factor^2), ....$ Internally decides whether or not to use `S-fn` or `next-S-fn` to generate successive estimates.
(left-integral f a b)(left-integral f a b opts)Returns an estimate of the integral of f across the closed-open interval $a,
b$ using a left-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate.
Optionally accepts opts, a dict of optional arguments. All of these get
passed on to us/seq-limit to configure convergence checking.
See left-sequence for information on the optional args in opts that
customize this function's behavior.
Returns an estimate of the integral of `f` across the closed-open interval $a, b$ using a left-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate. Optionally accepts `opts`, a dict of optional arguments. All of these get passed on to `us/seq-limit` to configure convergence checking. See `left-sequence` for information on the optional args in `opts` that customize this function's behavior.
(left-sequence f a b)(left-sequence f a b opts)Returns a (lazy) sequence of successively refined estimates of the integral of
f over the closed-open interval $a, b$ by taking left-Riemann sums.
:n: If n is a number, returns estimates with $n, 2n, 4n, ...$ slices,
geometrically increasing by a factor of 2 with each estimate.
If n is a sequence, the resulting sequence will hold an estimate for each
integer number of slices in that sequence.
:accelerate?: if supplied (and n is a number), attempts to accelerate
convergence using Richardson extrapolation. If n is a sequence this option
is ignored.
Returns a (lazy) sequence of successively refined estimates of the integral of `f` over the closed-open interval $a, b$ by taking left-Riemann sums. ## Optional Arguments `:n`: If `n` is a number, returns estimates with $n, 2n, 4n, ...$ slices, geometrically increasing by a factor of 2 with each estimate. If `n` is a sequence, the resulting sequence will hold an estimate for each integer number of slices in that sequence. `:accelerate?`: if supplied (and `n` is a number), attempts to accelerate convergence using Richardson extrapolation. If `n` is a sequence this option is ignored.
(left-sequence** f a b)(left-sequence** f a b n0)Returns a (lazy) sequence of successively refined estimates of the integral of
f over the closed-open interval $a, b$ by taking left-Riemann sums with
n0, 2n0, 4n0, ...
slices.
Returns a (lazy) sequence of successively refined estimates of the integral of `f` over the closed-open interval $a, b$ by taking left-Riemann sums with n0, 2n0, 4n0, ... slices.
(lower-integral f a b)(lower-integral f a b opts)Returns an estimate of the integral of f across the closed-open interval $a,
b$ using a lower-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate.
Optionally accepts opts, a dict of optional arguments. All of these get
passed on to us/seq-limit to configure convergence checking.
See lower-sequence for information on the optional args in opts that
customize this function's behavior.
Returns an estimate of the integral of `f` across the closed-open interval $a, b$ using a lower-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate. Optionally accepts `opts`, a dict of optional arguments. All of these get passed on to `us/seq-limit` to configure convergence checking. See `lower-sequence` for information on the optional args in `opts` that customize this function's behavior.
(lower-sequence f a b)(lower-sequence f a b {:keys [n] :or {n 1} :as opts})Returns a (lazy) sequence of successively refined estimates of the integral of
f over the closed interval $(a, b)$ by taking lower-Riemann sums.
:n: If n is a number, returns estimates with $n, 2n, 4n, ...$ slices,
geometrically increasing by a factor of 2 with each estimate.
If n is a sequence, the resulting sequence will hold an estimate for each
integer number of slices in that sequence.
:accelerate?: if supplied (and n is a number), attempts to accelerate
convergence using Richardson extrapolation. If n is a sequence this option
is ignored.
Returns a (lazy) sequence of successively refined estimates of the integral of `f` over the closed interval $(a, b)$ by taking lower-Riemann sums. ## Optional Arguments `:n`: If `n` is a number, returns estimates with $n, 2n, 4n, ...$ slices, geometrically increasing by a factor of 2 with each estimate. If `n` is a sequence, the resulting sequence will hold an estimate for each integer number of slices in that sequence. `:accelerate?`: if supplied (and `n` is a number), attempts to accelerate convergence using Richardson extrapolation. If `n` is a sequence this option is ignored.
(midpoint-sum f a b)Returns a function of n, some number of slices of the total integration
range, that returns an estimate for the definite integral of $f$ over the
range $(a, b)$ using midpoint estimates.
Returns a function of `n`, some number of slices of the total integration range, that returns an estimate for the definite integral of $f$ over the range $(a, b)$ using midpoint estimates.
(right-integral f a b)(right-integral f a b opts)Returns an estimate of the integral of f across the closed-open interval $a,
b$ using a right-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate.
Optionally accepts opts, a dict of optional arguments. All of these get
passed on to us/seq-limit to configure convergence checking.
See right-sequence for information on the optional args in opts that
customize this function's behavior.
Returns an estimate of the integral of `f` across the closed-open interval $a, b$ using a right-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate. Optionally accepts `opts`, a dict of optional arguments. All of these get passed on to `us/seq-limit` to configure convergence checking. See `right-sequence` for information on the optional args in `opts` that customize this function's behavior.
(right-sequence f a b)(right-sequence f a b opts)Returns a (lazy) sequence of successively refined estimates of the integral of
f over the closed-open interval $a, b$ by taking right-Riemann sums.
:n: If n is a number, returns estimates with $n, 2n, 4n, ...$ slices,
geometrically increasing by a factor of 2 with each estimate.
If n is a sequence, the resulting sequence will hold an estimate for each
integer number of slices in that sequence.
:accelerate?: if supplied (and n is a number), attempts to accelerate
convergence using Richardson extrapolation. If n is a sequence this option
is ignored.
Returns a (lazy) sequence of successively refined estimates of the integral of `f` over the closed-open interval $a, b$ by taking right-Riemann sums. ## Optional Arguments `:n`: If `n` is a number, returns estimates with $n, 2n, 4n, ...$ slices, geometrically increasing by a factor of 2 with each estimate. If `n` is a sequence, the resulting sequence will hold an estimate for each integer number of slices in that sequence. `:accelerate?`: if supplied (and `n` is a number), attempts to accelerate convergence using Richardson extrapolation. If `n` is a sequence this option is ignored.
(Sn->S2n f a b)Returns a function of:
Sn: a sum estimate for n partitions, andn: the number of partitionsAnd returns a new estimate for $S_{2n}$ by sampling the midpoints of each slice. This incremental update rule is valid for left and right Riemann sums, as well as the midpoint method.
Returns a function of:
- `Sn`: a sum estimate for `n` partitions, and
- `n`: the number of partitions
And returns a new estimate for $S_{2n}$ by sampling the midpoints of each
slice. This incremental update rule is valid for left and right Riemann sums,
as well as the midpoint method.(upper-integral f a b)(upper-integral f a b opts)Returns an estimate of the integral of f across the closed-open interval $a,
b$ using an upper-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate.
Optionally accepts opts, a dict of optional arguments. All of these get
passed on to us/seq-limit to configure convergence checking.
See upper-sequence for information on the optional args in opts that
customize this function's behavior.
Returns an estimate of the integral of `f` across the closed-open interval $a, b$ using an upper-Riemann sum with $1, 2, 4 ... 2^n$ windows for each estimate. Optionally accepts `opts`, a dict of optional arguments. All of these get passed on to `us/seq-limit` to configure convergence checking. See `upper-sequence` for information on the optional args in `opts` that customize this function's behavior.
(upper-sequence f a b)(upper-sequence f a b {:keys [n] :or {n 1} :as opts})Returns a (lazy) sequence of successively refined estimates of the integral of
f over the closed interval $(a, b)$ by taking upper-Riemann sums.
:n: If n is a number, returns estimates with $n, 2n, 4n, ...$ slices,
geometrically increasing by a factor of 2 with each estimate.
If n is a sequence, the resulting sequence will hold an estimate for each
integer number of slices in that sequence.
:accelerate?: if supplied (and n is a number), attempts to accelerate
convergence using Richardson extrapolation. If n is a sequence this option
is ignored.
Returns a (lazy) sequence of successively refined estimates of the integral of `f` over the closed interval $(a, b)$ by taking upper-Riemann sums. ## Optional Arguments `:n`: If `n` is a number, returns estimates with $n, 2n, 4n, ...$ slices, geometrically increasing by a factor of 2 with each estimate. If `n` is a sequence, the resulting sequence will hold an estimate for each integer number of slices in that sequence. `:accelerate?`: if supplied (and `n` is a number), attempts to accelerate convergence using Richardson extrapolation. If `n` is a sequence this option is ignored.
(windowed-sum area-fn a b)Takes:
area-fn, a function of the left and right endpoints of some integration
slicea and band returns a function of n, the number of slices to use for an integration
estimate.
area-fn should return an estimate of the area under some curve between the
l and r bounds it receives.
Takes: - `area-fn`, a function of the left and right endpoints of some integration slice - definite integration bounds `a` and `b` and returns a function of `n`, the number of slices to use for an integration estimate. `area-fn` should return an estimate of the area under some curve between the `l` and `r` bounds it receives.
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