Mathlib.Analysis.Certified­Contour­Interval­Arithmetic

For a rational interval and a real number, the containment relation holds precisely when the real number is no smaller than the interval's lower endpoint and no larger than its upper endpoint, after the rational endpoints are viewed as real numbers.

For a rational interval, its width is the upper rational endpoint minus the lower rational endpoint.

For a rational interval, its maximum endpoint magnitude is the larger of the absolute values of its two rational endpoints.

For two rational intervals, the subinterval relation holds precisely when the second interval's lower endpoint is no greater than the first interval's lower endpoint and the first interval's upper endpoint is no greater than the second interval's upper endpoint.

For a rational number, the point interval at that number is the rational interval whose lower endpoint is that number, upper endpoint is that number, and endpoints are ordered because they are equal.

For two rational intervals, their rational interval hull has the smaller lower endpoint, the larger upper endpoint, and these endpoints in weakly increasing order.

For two rational intervals, their interval sum has lower endpoint equal to the sum of the lower endpoints, upper endpoint equal to the sum of the upper endpoints, and valid endpoint order.

For a rational interval, its interval negation has lower endpoint equal to the negative of the original upper endpoint, upper endpoint equal to the negative of the original lower endpoint, and valid endpoint order.

For two rational intervals, their interval difference is the interval sum of the first interval and the interval negation of the second.

For two rational intervals, their interval product has lower endpoint equal to the least of the four products of one endpoint from each interval, upper endpoint equal to the greatest of those four products, and valid endpoint order.

For a rational interval, the property of being separated from zero holds precisely when the interval lies strictly below zero or strictly above zero.

For a rational interval that is separated from zero, its interval reciprocal has lower endpoint equal to the reciprocal of the original upper endpoint, upper endpoint equal to the reciprocal of the original lower endpoint, and valid endpoint order.

For a numerator rational interval and a denominator rational interval whose denominator interval is separated from zero, their interval quotient is the interval product of the numerator interval and the interval reciprocal of the denominator interval.

For a rational interval, its natural-power interval sequence assigns the point interval at one to exponent zero and the interval product of the preceding power and the original interval to each positive exponent.

For two rational intervals, their conditional tightening is their exact intersection when its lower endpoint does not exceed its upper endpoint, and the first interval otherwise.

For a rational interval, a rational widening amount that is nonnegative, the expanded interval has lower endpoint equal to the original lower endpoint minus the widening amount, upper endpoint equal to the original upper endpoint plus the widening amount, and valid endpoint order.

A positive rational number is a rational number that is strictly greater than zero.

For a certified real number and a requested positive rational tolerance, its refinement is the rational interval returned at the precision selected by that certificate's modulus for the requested tolerance.

For a rational number, its certified real name denotes that rational number and uses its degenerate one-point rational interval at every precision.

For a certified real number, its certified negation denotes the negative of the real number named by the certificate, negates every rational interval enclosure, and retains the same precision-selection rule.

For two certified real numbers, their certified sum denotes the sum of their named real numbers, adds their rational interval enclosures at each common precision, and selects a precision sufficient to make each input enclosure no wider than half the requested tolerance.

For two certified real numbers, their certified difference is their certified sum after negating the second certificate; it therefore denotes the first named real number minus the second.

Given a rational rectangle and a complex number, rectangle membership means that both the real and imaginary coordinates of the number lie in their respective rational intervals.

Given two rational rectangles in the complex plane, the refinement relation holds exactly when each coordinate interval of the first is contained in the corresponding coordinate interval of the second.

For two rational rectangles in the complex plane, their sum rectangle is obtained by adding their real-coordinate intervals and their imaginary-coordinate intervals separately.

For a rational scalar and a rational rectangle in the complex plane, the scalar-multiple rectangle is obtained by multiplying each coordinate interval by that scalar.

For a rational rectangle in the complex plane, a rational error allowance, and the condition that this allowance is nonnegative, the expanded rectangle widens each coordinate interval by that allowance.

The zero rectangle is the rational rectangle whose real and imaginary coordinate intervals are both the singleton interval containing zero.

For a mesh size and a node index, the mesh point is the real number $k/n$. It is a mesh point for positive mesh size; for $n=0$ every point is zero by the division convention.

For a complex center, a real radius, and a real parameter, the circle point is $c+r\exp(2\pi u i)$.

For a real radius and a real parameter, the circle tangent is $2\pi i r\exp(2\pi u i)$.

For a complex-valued function, a complex center, a real radius, and a real parameter, the parameterized contour integrand is the value of the function at the corresponding circle point multiplied by the circle tangent there.

For a complex-valued function, a complex center, and a real radius, the circle contour integral is the integral from $0$ to $1$ of the parameterized contour integrand. No integrability is required: for an integrand that is not interval integrable the integral is zero by convention, so this is the contour integral only when the parameterized integrand is integrable.

Given a sequence of rational intervals, the minimum lower endpoint up to a terminal node index is computed recursively: at index zero it is the lower endpoint of the initial interval, and at each successor index it is the smaller of the preceding result and the new lower endpoint.

Given a sequence of rational intervals, the minimum upper endpoint up to a terminal node index is computed recursively: at index zero it is the upper endpoint of the initial interval, and at each successor index it is the smaller of the preceding result and the new upper endpoint.

Given a sequence of rational intervals, the maximum lower endpoint up to a terminal node index is computed recursively: at index zero it is the lower endpoint of the initial interval, and at each successor index it is the larger of the preceding result and the new lower endpoint.

Given a sequence of rational intervals, the maximum upper endpoint up to a terminal node index is computed recursively: at index zero it is the upper endpoint of the initial interval, and at each successor index it is the larger of the preceding result and the new upper endpoint.

Given a sequence of rational node intervals, a rational Lipschitz constant, the condition that the constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the infimum enclosure has lower endpoint equal to the minimum node lower endpoint minus $L/n$ and upper endpoint equal to the minimum node upper endpoint.

Given a sequence of rational node intervals, a rational Lipschitz constant, the condition that the constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the supremum enclosure has lower endpoint equal to the maximum node lower endpoint and upper endpoint equal to the maximum node upper endpoint plus $L/n$.

For a rational exponential argument, the exponential scaling factor is the larger of one and the absolute value of the argument's numerator.

For a rational exponential argument, the reduced exponential argument is the argument divided by its exponential scaling factor.

For a rational number and a nonnegative polynomial degree, the exponential Taylor partial sum is the sum of the terms $q^k/k!$ for every integer $k$ from zero through $n$.

For a rational number and a nonnegative polynomial degree, the exponential remainder radius is $|q|^{n+1}(n+2)/((n+1)!(n+1))$.

For a rational number and a nonnegative polynomial degree, the reduced exponential enclosure has lower endpoint equal to the Taylor partial sum minus the remainder radius, upper endpoint equal to the Taylor partial sum plus the remainder radius, and valid endpoint order.

For a rational exponential argument and a nonnegative polynomial degree, the raw exponential enclosure is the natural-power interval obtained by raising the reduced exponential enclosure to the exponential scaling factor.

For a rational exponential argument, the scalar exponential enclosure sequence assigns the raw exponential enclosure at degree zero to index zero and the conditional tightening of the preceding enclosure with the next raw enclosure to each positive index.

For a rational exponential argument and a strictly positive rational target width, the exponential precision index is the product of one plus the target-width denominator and two copies of one plus the exponential scaling factor.

For a rational number, the certified-real representation of its exponential has value equal to the real exponential of that rational number, approximating intervals given by the scalar exponential enclosure sequence, nested approximations, containment of the exact value, the exponential precision index as its modulus, and the corresponding target-width guarantee.

Given a sequence of complex rational rectangles, the unscaled trapezoidal rectangle sum at a nonnegative index is computed recursively: at zero it is the zero rectangle, and at each successor it adds the two rectangles at the new cell's endpoints to the preceding sum.

For a sequence of complex rational rectangles and a mesh size, the trapezoidal enclosure is the unscaled trapezoidal rectangle sum multiplied by $1/(2n)$. It is meaningful for positive mesh size, as required by the integral enclosure built from it; at mesh size zero the scale factor is zero by the division convention.

Given a sequence of complex rational node rectangles, a rational Lipschitz constant, the condition that this constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the integral enclosure widens the trapezoidal enclosure by $L/(2n)$ in both coordinates.

The concrete certified interval-arithmetic package provides outwardly sound rational interval operations for addition, subtraction, multiplication, division away from zero, exponentiation, logarithms and real powers on their positive domains, together with certified-real refinement of any requested positive rational precision.

For a nonnegative integer and a Boolean-valued family indexed by the $n+1$ indices from zero through $n, the least-successful-index rule returns the smallest index whose Boolean value is true when such an index exists, and index zero when none exists.

For a nonnegative integer, a family of $n+1$ rational candidate intervals, and a strictly positive rational width tolerance, the finite-refinement interval is the candidate at the least index whose width is at most the tolerance, or the zeroth candidate when no candidate meets that condition.

For a nonnegative integer and a real-valued score for each of the $n+1$ indices from zero through $n, the least-score index is obtained by forming the finite set of pairs consisting of each score and its index, ordered first by score and then by index, certifying that this finite set is nonempty, and returning the index in its least pair.

For a rational number, the logarithm coordinate is $(q-1)/(q+1)$. It is the atanh coordinate of the logarithm for positive $q$, the regime used by the logarithm enclosures; at $q=-1$ the denominator vanishes and the value is zero by the division convention.

For a rational number and a nonnegative polynomial degree, the logarithm Taylor partial sum is twice the sum, for every integer $k$ from zero through $n$, of the $(2k+1)$st power of the logarithm coordinate divided by $2k+1$. It approximates the logarithm of $q$ only for positive $q$; no positivity is imposed by the definition itself.

For a rational number and a nonnegative polynomial degree, the logarithm remainder bound is $2|z|^{2n+3}/((2n+3)(1-|z|^2))$, where $z=(q-1)/(q+1)$. It bounds the series tail only for positive $q$, where $|z|<1$; at $q=0$ (where $|z|=1$) and at $q=-1$ the formula divides by zero and gives zero by the division convention.

For a rational number that is strictly positive and a nonnegative polynomial degree, the raw logarithm enclosure has lower endpoint equal to the logarithm partial sum minus its remainder bound, upper endpoint equal to the logarithm partial sum plus its remainder bound, and valid endpoint order.

For a strictly positive rational number, the scalar logarithm enclosure sequence assigns the raw logarithm enclosure at degree zero to index zero and the conditional tightening of the preceding enclosure with the next raw enclosure to each positive index.

For a rational number and a strictly positive rational target width, the logarithm precision index is the product of one plus the target-width denominator and two copies of one plus the sum of the absolute numerator and denominator of the rational number.

For a rational number that is strictly positive, the certified-real representation of its logarithm has value equal to the real logarithm of that rational number, approximating intervals given by the scalar logarithm enclosure sequence, nested approximations, containment of the exact value, the logarithm precision index as its modulus, and the corresponding target-width guarantee.