如何防止浮点数不精确影响numpy.arange? [英] How to prevent float imprecision from affecting numpy.arange?
问题描述
因为 numpy.arange()使用ceil((stop - start)/step)确定项目数，小的浮点不精确度(stop = .400000001)可以在列表中添加意外的值.
Because numpy.arange() uses ceil((stop - start)/step) to determine the number of items, a small float imprecision (stop = .400000001) can add an unintended value to the list.
第一种情况不包括停止点(预定)
The first case does not include the stop point (intended)
>>> print(np.arange(.1,.3,.1))
[0.1 0.2]
第二种情况包括停止点(不希望)
The second case includes the stop point (not intended)
>>> print(np.arange(.1,.4,.1))
[0.1 0.2 0.3 0.4]
numpy.linspace()修复了此问题问题，np.linspace(.1,.4-.1,3).但需要您知道步骤数. np.linspace(start,stop-step,np.ceil((stop-step)/step))导致相同的无能.
numpy.linspace() fixes this problem, np.linspace(.1,.4-.1,3). but requires you know the number of steps. np.linspace(start,stop-step,np.ceil((stop-step)/step)) leads to the same incosistencies.
如何在不知道范围内元素数量的情况下生成可靠的float范围?
How can I generate a reliable float range without knowing the # of elements in the range?
考虑要生成精度未知的浮点索引的情况
Consider the case in which I want generate a float index of unknown precision
np.arange(2.00(...)001,2.00(...)021,.00(...)001)
推荐答案
您的目标是计算如果值是使用精确数学计算出的ceil((stop - start)/step).
Your goal is to calculate what ceil((stop - start)/step) would be if the values had been calculated with exact mathematics.
仅给出 浮点值start，stop和step是可能会发生一些舍入错误的运算结果，因此这是不可能的.四舍五入会删除信息，并且根本无法从信息不足的情况下创建信息.
This is impossible to do given only floating-point values of start, stop, and step that are the results of operations in which some rounding errors may have occurred. Rounding removes information, and there is simply no way to create information from lack of information.
因此，仅当您具有有关start，stop和step的其他信息时，此问题才可以解决.
Therefore, this problem is only solvable if you have additional information about start, stop, and step.
假设step是精确的，但是start和stop积累了一些受e0和e1限制的错误.也就是说，您知道start与理想数学值(在任一方向上)最多为e0，并且stop与理想数学值(在任一方向上)最多是e1.这样，(stop-start)/step的理想值可能会从其理想值开始在(stop-start-e0-e1)/step到(stop-start+e0+e1)/step的范围内.
Suppose step is exact, but start and stop have some accumulated errors bounded by e0 and e1. That is, you know start is at most e0 away from its ideal mathematical value (in either direction), and stop is at most e1 away from its ideal value (in either direction). Then the ideal value of (stop-start)/step could range from (stop-start-e0-e1)/step to (stop-start+e0+e1)/step away from its ideal value.
假设(stop-start-e0-e1)/step至(stop-start+e0+e1)/step之间存在整数.然后，仅根据start，stop和step的浮点值以及边界e0和
Suppose there is an integer between (stop-start-e0-e1)/step to (stop-start+e0+e1)/step. Then it is impossible to know whether the ideal ceil result should be the lesser integer or the greater just from the floating-point values of start, stop, and step and the bounds e0 and e1.
但是，从您给出的示例来看，理想的(stop-start)/step可能正好是整数，如(.4-.1)/.1所示.如果是这样，则任何非零的错误界限都可能导致错误间隔跨越整数，从而使该问题无法从到目前为止的信息中解决.
However, from the examples you have given, the ideal (stop-start)/step could be exactly an integer, as in (.4-.1)/.1. If so, any non-zero error bounds could result in the error interval straddling an integer, making the problem impossible to solve from the information we have so far.
因此，为了解决问题，您不仅必须具有关于错误的简单界限，还必须具有更多信息.例如，您必须知道(stop-start)/step恰好是整数或以其他方式量化.例如，如果您知道理想的步数计算将产生0.1的倍数，例如3.8、3.9、4.0、4.1或4.2，但绝不会是4.05，并且误差足够小，以至于浮动-点计算(stop-start)/step的最终误差小于.05，则可以将(stop-start)/step舍入为最接近的合格倍数，然后对其应用ceil.
Therefore, in order to solve the problem, you must have more information than just simple bounds on the errors. You must know, for example, that (stop-start)/step is exactly an integer or is otherwise quantized. For example, if you knew that the ideal calculation of the number of steps would produce a multiple of .1, such as 3.8, 3.9, 4.0, 4.1, or 4.2, but never 4.05, and the errors were sufficiently small that the floating-point calculation (stop-start)/step had a final error less than .05, then it would be possible to round (stop-start)/step to the nearest qualifying multiple and then to apply ceil to that.
如果您有这样的信息，则可以使用关于start，stop和step(例如的错误)的已知知识来更新问题.从十进制到浮点的单次转换的结果)和理想(stop-start)/step的可能值.如果您没有这样的信息，那就没有解决办法.
If you have such information, you can update the question with what you know about the errors in start, stop, and step (e.g., perhaps each of them is the result of a single conversion from decimal to floating-point) and the possible values of the ideal (stop-start)/step. If you do not have such information, there is no solution.
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