2026 年高考题：用 Taylor 级数前两项求方程 cosx=x 的近似解

lihp2020

其实我想说 牛顿迭代法  中 的 【cos(xn) -xn】  明显 这个几乎接近与 0  cos(xn) 按照计算机的计算规则 也是近似计算      

一切都是近似 计算 一个精确到某某的数  准确吗？  

elim

lihp2020 发表于 2026-6-18 01:52
其实我想说 牛顿迭代法  中 的 【cos(xn) -xn】  明显 这个几乎接近与 0  cos(xn) 按照计算机的计算规则 也 ...

准确与否这件事在实际计算中变成与理论精确值的接近程度的问题. 我们
也给不出 cos(x) 的准确值。但是我们要求系统给出 cos(x) 值的足够准
(例如 100，或一千位有效数字), 然后我们的迭代又可以继续修正已经有
的近似值，直到迭代前后的差别已经超出所要求的误差精度。

我们的迭代结果
\(x = 0.7390851332151606416553120876738734040134117589\ldots\)
与 Mathematica 的结果相符:

ataorj

cosx=x就一个“变量x”，用cosx得到新的x，直到反转新x=cosx,记为X
这里暂时只考虑0-pi/4，X显然唯一，x若>X则新x<X,相反略。
这样，连续得到的x总是小大小大不停变化的，任何一个小和一个大的开始起的连续共有数字就肯定确认是X的前面起的有效数字。这种已知共有最多的大小对就能确认出最大的精度值。

ataorj

x>X,则cosx<X则coscosx>x且coscosx>X，...，可见大值越来越大，小值越来越小，都是远离X，所以初始不能让x>X。
显然只要初始x<X则精度越来越高。

ataorj

我前面发言是脑中有个单位圆,分析∠p<∠X<∠q时的余弦变化,
没考虑本题要求余弦和角度角色转换,所以有问题和错误.
如图y2y3对应我说的一小一大,共用一条垂线,B左右下的
大小关系其实是反转的.另外,'大'并非一直>X,A下时的大就不行.
至于'共有数字'和精度关系这里仍然应该存疑.

ataorj

好像论坛有问题,补充图片

ataorj

再次试验补充图片...

ataorj

三次pc上发图不行，这次手机试

ataorj

上图有误。     

王守恩

elim 发表于 2026-6-19 08:12
准确与否这件事在实际计算中变成与理论精确值的接近程度的问题. 我们
也给不出 cos(x) 的准确值。但是 ...

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}+\cdots}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots}=?\)        这个 ？有可能到达 e 吗 ？

{001, 1.00000}, {002, 1.20711}, {003, 1.33333}, {004, 1.42084}, {005, 1.48635}, {006, 1.53794}, {007, 1.58004}, {008, 1.61532}, {009, 1.64551}, {010, 1.67176},
{011, 1.69489}, {012, 1.71550}, {013, 1.73403}, {014, 1.75083}, {015, 1.76616}, {016, 1.78023}, {017, 1.79321}, {018, 1.80525}, {019, 1.81645}, {020, 1.82692},
{021, 1.83674}, {022, 1.84597}, {023, 1.85468}, {024, 1.86291}, {025, 1.87070}, {026, 1.87811}, {027, 1.88515}, {028, 1.89187}, {029, 1.89828}, {030, 1.90441},
{031, 1.91029}, {032, 1.91592}, {033, 1.92133}, {034, 1.92653}, {035, 1.93154}, {036, 1.93636}, {037, 1.94102}, {038, 1.94552}, {039, 1.94986}, {040, 1.95406},
{041, 1.95813}, {042, 1.96208}, {043, 1.96590}, {044, 1.96961}, {045, 1.97321}, {046, 1.97671}, {047, 1.98012}, {048, 1.98343}, {049, 1.98665}, {050, 1.98979},
{051, 1.99285}, {052, 1.99584}, {053, 1.99874}, {054, 2.00158}, {055, 2.00436}, {056, 2.00707}, {057, 2.00971}, {058, 2.01230}, {059, 2.01483}, {060, 2.01731},
{061, 2.01973}, {062, 2.02210}, {063, 2.02442}, {064, 2.02670}, {065, 2.02893}, {066, 2.03112}, {067, 2.03326}, {068, 2.03537}, {069, 2.03743}, {070, 2.03946},
{071, 2.04145}, {072, 2.04340}, {073, 2.04532}, {074, 2.04721}, {075, 2.04906}, {076, 2.05088}, {077, 2.05267}, {078, 2.05443}, {079, 2.05616}, {080, 2.05787},
{081, 2.05955}, {082, 2.06120}, {083, 2.06282}, {084, 2.06442}, {085, 2.06600}, {086, 2.06754}, {087, 2.06908}, {088, 2.07058}, {089, 2.07207}, {090, 2.07353},
{091, 2.07497}, {092, 2.07640}, {093, 2.07780}, {094, 2.07918}, {095, 2.08055}, {096, 2.08189}, {097, 2.08322}, {098, 2.08453}, {099, 2.08583}, {100, 2.08711},
{101, 2.08837}, {102, 2.08961}, {103, 2.09084}, {104, 2.09206}, {105, 2.09325}, {106, 2.09444}, {107, 2.09561}, {108, 2.09677}, {109, 2.09791}, {110, 2.09904},
{111, 2.10016}, {112, 2.10126}, {113, 2.10235}, {114, 2.10343}, {115, 2.10450}, {116, 2.10555}, {117, 2.10660}, {118, 2.10763}, {119, 2.10865}, {120, 2.10966},
{121, 2.11066}, {122, 2.11165}, {123, 2.11262}, {124, 2.11359}, {125, 2.11455}, {126, 2.11550}, {127, 2.11644}, {128, 2.11737}, {129, 2.11829}, {130, 2.11920},
{131, 2.12010}, {132, 2.12145}, {133, 2.12188}, {134, 2.12276}, {135, 2.12363}, {136, 2.12449}, {137, 2.12534}, {138, 2.12618}, {139, 2.12702}, {140, 2.12785},
{141, 2.12867}, {142, 2.12948}, {143, 2.13029}, {144, 2.13109}, {145, 2.13188}, {146, 2.13267}, {147, 2.13344}, {148, 2.13422}, {149, 2.13498}, {150, 2.13574},
{151, 2.13649}, {152, 2.13724}, {153, 2.13798}, {154, 2.13871}, {155, 2.13944}, {156, 2.14016}, {157, 2.14087}, {158, 2.14158}, {159, 2.14229}, {160, 2.14299},
{161, 2.14368}, {162, 2.14437}, {163, 2.14505}, {164, 2.14572}, {165, 2.14640}, {166, 2.14707}, {167, 2.14772}, {168, 2.14838}, {169, 2.14903}, {170, 2.14968},
{171, 2.15032}, {172, 2.15095}, {173, 2.15159}, {174, 2.15221}, {175, 2.15283}, {176, 2.15346}, {177, 2.15407}, {178, 2.15468}, {179, 2.15528}, {180, 2.15588}}