CenteredInterval[x,dx]
对于实数 x 和 dx 给出一个包含实数区间
的中心区间.
CenteredInterval[x+ y,dx+ dy]
给出一个包含复数矩形
的中心区间.
对于近似数 c 给出一个中心区间,该区间包含 c 的误差范围内的所有值.
CenteredInterval
CenteredInterval[x,dx]
对于实数 x 和 dx 给出一个包含实数区间
的中心区间.
CenteredInterval[x+ y,dx+ dy]
给出一个包含复数矩形
的中心区间.
对于近似数 c 给出一个中心区间,该区间包含 c 的误差范围内的所有值.
更多信息
- 中心区间也称为中心半径或中间半径区间.
- CenteredInterval 通常用于获得通过数值计算累积的误差的确信范围. 给定函数所有参数的误差范围,中心区间计算为函数值中的误差提供了可靠的范围.
- CenteredInterval[…] 给出中心为
、半径为
的中心区间对象 Δ,其中
和
是具有两个分母幂的高斯有理数. 如果
和
为实数,则 Δ 表示实数区间
;否则 Δ 表示复数矩形
. - 算术运算和许多数学函数使用中心区间参数. 对于任何 ai∈Δi,f[Δ1,…,Δn] 生成包含 f[a1,…,an] 的中心区间对象 Δ.
- IntervalMemberQ 可用于确定区间成员或区间之间的包含.
- 只要给定不相交的区间,Equal 和 Less 之类的关系运算符就会产生明确的 True 或 False 结果.
- 在 StandardForm 和相关格式中,CenteredInterval 对象以省略形式打印,仅显示中心和半径的近似值.
- Normal 将 CenteredInterval 对象转换为精度与半径对应的任意精度数.
- Information[CenteredInterval[…], prop] 给出中心半径区间的属性 prop. 可以指定以下属性:
-
"Center" 区间的中心 "Radius" 区间的半径 "Bounds" 区间内值的界限 - 线性代数运算,如 Det、Inverse、LinearSolve 和 Eigensystem,可用于有 CenteredInterval 项的矩阵.
范例
打开所有单元 关闭所有单元基本范例 (3)
CenteredInterval[2, 10 ^ -10]% Log[%] + Sin[% ^ 2 + 1]Normal[%]CenteredInterval[2 + 3I, 10 ^ -10 + 5 10 ^ -9 I]E ^ Gamma[%]Information[%, {"Center", "Radius"}]bds = Information[Exp[CenteredInterval[0, 1 + I]], "Bounds"]reg = Region[Style[TransformedRegion[Rectangle[{-1, -1}, {1, 1}], Function[{a, b}, {E^a Cos[b], E^a Sin[b]}]], Red]];Show[{Graphics[{Yellow, Rectangle@@(ReIm /@ bds)}], reg}]范围 (27)
构造中心-半径区间 (7)
CenteredInterval[2 / 3, 10 ^ -10]CenteredInterval[2 + 3I, 1 + I]CenteredInterval[1.23`20]CenteredInterval[1.23`20 + 4.56`30 I]CenteredInterval[0``7]将有界 Interval 对象转换为中心区间:
CenteredInterval[Interval[{1, 2}]]CenteredInterval[123 / 2 ^ 12]CenteredInterval[1 / 2 + 3 / 4I]CenteredInterval[1 / 3]CenteredInterval[E + I Pi]非零机器精度数被视为具有 $MachinePrecision 精确数字的数:
CenteredInterval[9.87]CenteredInterval[0.]区间算术 (5)
{a, b} = {CenteredInterval[2, 1 / 100], CenteredInterval[3 + I / 4, (1 + I) / 20]};{a + b, a b, a ^ b}Sin[CenteredInterval[1.23`4]]BesselJ[CenteredInterval[1, 1 / 1000], CenteredInterval[2, 1 / 1000]]a = CenteredInterval[1.234`10];a + PiBeta[2`15, a] + 3 / 4Log[#]Sin[#]&[CenteredInterval[20, 10]]Plot[Log[x]Sin[x], {x, 10, 30}, Epilog -> {Green, Opacity[0.25], Rectangle[{10, -#}, {30, #}]&[Information[%, "Radius"]]}]如果值集是无界的,则区间算术运算返回 Indeterminate:
1 / CenteredInterval[0, 1]区间属性 (5)
int = CenteredInterval[1.23`20]Information[int, {"Center", "Radius"}]Information[int, "Bounds"]N[%, 21]Normal[int]int = CenteredInterval[N[ArcSin[2], 10]]Information[int, {"Center", "Radius"}]Information[int, "Bounds"]N[%, 11]Normal[int]int = CenteredInterval[2, 1 / 2]IntervalMemberQ[int, 9 / 4]IntervalMemberQ[int, 5 / 4]int1 = CenteredInterval[0, 1 + I]int2 = CenteredInterval[1 / 3, 1 / 3 + 1 / 3I]IntervalMemberQ[int1, int2]IntervalMemberQ[int2, int1]Graphics[{{Red, Rectangle@@ReIm[Information[int1, "Bounds"]]}, {Yellow, Rectangle@@ReIm[Information[int2, "Bounds"]]}}]int1 = CenteredInterval[1, 3 / 4];
int2 = CenteredInterval[2, 3 / 4];
int3 = CenteredInterval[3, 3 / 4];int1 < int3int1 < int2使用 IntervalIntersection 计算交点:
IntervalIntersection[int1, int2]空区间表示为 Interval[]:
IntervalIntersection[int1, int3]线性代数 (10)
CenteredInterval 矩阵的乘积:
m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}];
n = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 2}, WorkingPrecision -> 10], {2}];
(mn = m.n)//MatrixFormranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixForm(nrep = Map[ranrep, n, {2}])//MatrixFormMapThread[IntervalMemberQ, {mn, mrep.nrep}, 2]//MatrixForm将 CenteredInterval 矩阵提升到整数幂:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixForm(mpow = MatrixPower[m, 17])//MatrixFormranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixForm验证 mpow 包含了 MatrixPower[mrep,17]:
MapThread[IntervalMemberQ, {mpow, MatrixPower[mrep, 17]}, 2]//MatrixFormCenteredInterval 矩阵的指数:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixForm(mexp = MatrixExp[m])//MatrixFormranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixFormMapThread[IntervalMemberQ, {mexp, MatrixExp[mrep]}, 2]//MatrixFormCenteredInterval 矩阵的行列式:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixFormmdet = Det[m]ranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixFormIntervalMemberQ[mdet, Det[mrep]]CenteredInterval 矩阵的逆矩阵:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixForm(minv = Inverse[m])//MatrixFormranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixFormMapThread[IntervalMemberQ, {minv, Inverse[mrep]}, 2]//MatrixForm求解 CenteredInterval 矩阵中
:
m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}];
b = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 2}, WorkingPrecision -> 10], {2}];
(sol = LinearSolve[m, b])//MatrixFormranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixForm(brep = Map[ranrep, b, {2}])//MatrixForm验证 sol 包含 LinearSolve[mrep,brep]:
MapThread[IntervalMemberQ, {sol, LinearSolve[mrep, brep]}, 2]//MatrixFormCenteredInterval 矩阵的特征系统:
SeedRandom[777];(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixForm{vals, vecs} = Eigensystem[m]ranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixForm{rvals, rvecs} = Eigensystem[mrep]验证矢量重新排序和缩放后,vals 包含 rvals,vecs 包含 rvecs:
MapThread[IntervalMemberQ, {vals, rvals[[{3, 1, 2}]]}]MapThread[IntervalMemberQ, {vecs, rvecs[[{3, 1, 2}]] * (Last[#]["Center"]& /@ vecs)}, 2]CenteredInterval 矩阵的 LU 分解:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixFormMatrixForm /@ ({l, u, p, c} = LUDecomposition[m])MapThread[IntervalMemberQ, {l.u, p.m}, 2]//MatrixForm实对称正定 CenteredInterval 矩阵的 Cholesky 分解:
rm = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}];
(m = rm.Transpose[rm])//MatrixForm(q = CholeskyDecomposition[m])//MatrixFormMapThread[IntervalMemberQ, {Transpose[q].q, m}, 2]//MatrixFormCenteredInterval 矩阵的特征多项式:
(m = Map[CenteredInterval, RandomReal[{-10, 10}, {3, 3}, WorkingPrecision -> 10], {2}])//MatrixFormp = CharacteristicPolynomial[m, x]ranrep[e_CenteredInterval] := e["Center"] + RandomInteger[{-1000, 1000}] / 1000 e["Radius"]
(mrep = Map[ranrep, m, {2}])//MatrixFormMapThread[IntervalMemberQ, {CoefficientList[p, x], CoefficientList[CharacteristicPolynomial[mrep, x], x]}]属性和关系 (2)
a = CenteredInterval[1, 1 / 10];a ^ 2CenteredInterval[Normal[a] ^ 2]Interval 表示通过指定端点给出的实际区间:
Interval[{1, 2}]将区间转换为 CenteredInterval 表示:
CenteredInterval[%]Interval[Information[%, "Bounds"]]Interval[{0, 1 / 3}]CenteredInterval[%]Interval[Information[%, "Bounds"]]N[%[[1]] - %%%[[1]]]可能存在的问题 (1)
只有有界区间可以表示为 CenteredInterval:
Tan[CenteredInterval[N[Pi / 2, 20]]]Interval 表示允许无界区间:
Tan[Interval[N[Pi / 2, 20]]]文本
Wolfram Research (2021),CenteredInterval,Wolfram 语言函数,https://reference.wolfram.com/language/ref/CenteredInterval.html.
CMS
Wolfram 语言. 2021. "CenteredInterval." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/CenteredInterval.html.
APA
Wolfram 语言. (2021). CenteredInterval. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/CenteredInterval.html 年
BibTeX
@misc{reference.wolfram_2026_centeredinterval, author="Wolfram Research", title="{CenteredInterval}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CenteredInterval.html}", note=[Accessed: 18-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_centeredinterval, organization={Wolfram Research}, title={CenteredInterval}, year={2021}, url={https://reference.wolfram.com/language/ref/CenteredInterval.html}, note=[Accessed: 18-June-2026]}