数值计算之求积分
数值计算之求积分
highflybird
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用LISP编写了一个求积分的程序：
里面采用了各种方法求积分和各种类型的积分。下面我把各种方法的源码贴出。
方法一： 勒让德-高斯积分法。
方法二：高斯-埃米尔特积分
方法三：高斯-埃米尔特积分
高斯-雅克比积分
;;;=============================================================
;;; 高斯-雅克比积分
;;; 功能: 计算 f(x)*((1-x)^a)*((1+x)^b)的积分(在区间-1..1上)
;;;=============================================================
(defun MATH::INT:Gauss-Jacobi (a b eps / ALF ARGS BET G N X Y flag g0)
(if (setq args (UTI:InputBox))
(progn
(setq flag T) ;是否进行迭代
(setq n 10)
(while (and (< n 100) flag)
(setq alf (car args))
(setq bet (cadr args))
(setq g 0)
(foreach w (MATH::INT:GetJacobiPolynomial n alf bet)
(setq x (car w))
(setq y (MATH::INT:func x))
(setq g (+ g (* (cdr w) y)))
)
(if (equal g g0 eps)
(setq flag nil g0 g)
(setq n (+ n n) g0 g)
)
)
)
)
)
;;;=============================================================
;;; 高斯-雅克比积分
;;; 功能: 计算高斯-雅克比积分系数项
;;;=============================================================
(defun MATH::INT:GetJacobiPolynomial (n alf bet /
xi wi A ALFBET AN B BN C EPS FI FLG I ITS
J MAXIT P1 P2 P3 PP R1 R2 R3 TEMP Z Z1)
(setq maxIT 40)
(setq eps 1e-15)
(setq alf (float alf))
(setq bet (float bet))
(setq i 0)
(repeat n ;for (i=0;i<n;i++) {
(cond
( (= i 0) ;if (i == 0) {
(setq an (/ alf n)) ;an=alf/n;
(setq bn (/ bet n)) ;bn=bet/n;
(setq r1 (* (1+ alf) (+ (/ 2.78 (+ 4.0 (* n n))) (/ (* 0.768 an) n)))) ;r1=(1.0+alf)*(2.78/(4.0+n*n)+0.768*an/n);
(setq r2 (+ 1.0 (* 1.48 n) (* 0.96 bn) (* 0.452 an an) (* 0.83 an bn))) ;r2=1.0+1.48*an+0.96*bn+0.452*an*an+0.83*an*bn;
(setq z (- 1 (/ r1 r2))) ;z=1.0-r1/r2;
)
( (= i 1) ;else if (i == 1)
(setq r1 (/ (+ 4.1 alf) (* (1+ alf) (1+ (* 0.156 alf))))) ;r1=(4.1+alf)/((1.0+alf)*(1.0+0.156*alf));
(setq r2 (1+ (/ (* 0.06 (- n 8.0) (1+ (* 0.12 alf))) n))) ;r2=1.0+0.06*(n-8.0)*(1.0+0.12*alf)/n;
(setq r3 (1+ (/ (* 0.012 bet (1+ (* 0.25 (abs alf)))) n))) ;r3=1.0+0.012*bet*(1.0+0.25*fabs(alf))/n;
(setq z (- z (* (- 1 z) r1 r2 r3))) ;z -= (1.0-z)*r1*r2*r3;
)
( (= i 2) ;else if (i == 2)
(setq r1 (/ (+ 1.67 (* 0.28 alf)) (1+ (* 0.37 alf)))) ;r1=(1.67+0.28*alf)/(1.0+0.37*alf);
(setq r2 (1+ (/ (* 0.22 (- n 8.0)) n))) ;r2=1.0+0.22*(n-8.0)/n;
(setq r3 (1+ (/ (* 8.0 bet) (* (+ 6.28 bet) n n)))) ;r3=1.0+8.0*bet/((6.28+bet)*n*n);
(setq z (- z (* (- (cdr (assoc 0 xi)) z) r1 r2 r3))) ;z -= (x[0]-z)*r1*r2*r3;
)
( (= i (- n 2)) ;else if (i == n-2)
(setq r1 (/ (1+ (* 0.235 bet)) (+ 0.766 (* 0.119 bet)))) ;r1=(1.0+0.37*bet)/(1.67+0.28*bet);
(setq r2 (/ 1.0 (1+ (/ (* 0.639 (- n 4.0)) (1+ (* 0.71 (- n 4.0))))))) ;r2=1.0/(1.0+0.639*(n-4.0)/(1.0+0.71*(n-4.0)));
(setq r3 (/ 1.0 (1+ (/ (* 20.0 alf) (* (+ 7.5 alf) n n))))) ;r3=1.0/(1.0+20.0*alf/((7.5+alf)*n*n));
(setq z (+ z (* (- z (cdr (assoc (- n 4) xi))) r1 r2 r3))) ;z += (z-x[n-4])*r1*r2*r3;
)
( (= i (1- n)) ;else if (i == n-1)
(setq r1 (/ (1+ (* 0.37 bet))(+ 1.67 (* 0.28 bet)))) ;r1=(1.0+0.37*bet)/(1.67+0.28*bet);
(setq r2 (/ 1.0 (1+ (/ (* 0.22 (- n 8.0)) n)))) ;r2=1.0/(1.0+0.22*(n-8.0)/n);
(setq r3 (/ 1.0 (1+ (/ (* 8.0 alf) (* (+ 6.28 alf) n n))))) ;r3=1.0/(1.0+8.0*alf/((6.28+alf)*n*n));
(setq z (+ z (* (- z (cdr (assoc (- n 3) xi))) r1 r2 r3))) ;z += (z-x[n-3])*r1*r2*r3;
)
(t ;else {
(setq z (* 3 (- (cdr (assoc (1- i) xi)) (cdr (assoc (- i 2) xi))))) ;z=3.0*x[i-1]-3.0*x[i-2]+x[i-3];
(setq z (+ z (cdr (assoc (- i 3) xi))))
)
)
(setq alfbet (+ alf bet)) ;alfbet=alf+bet;
(setq its 1)
(setq flg T)
(while (and flg (<= its maxIt)) ;for (its=1;its<=MAXIT;its++)
(setq temp (+ 2.0 alfbet)) ;temp=2.0+alfbet;
(setq p1 (/ (+ (- alf bet) (* temp z)) 2.0)) ;p1=(alf-bet+temp*z)/2.0;
(setq p2 1.0) ;p2=1.0;
(setq j 2)
(while (<= j n) ;for (j=2;j<=n;j++) {
(setq p3 p2) ;p3=p2;
(setq p2 p1) ;p2=p1;
(setq temp (+ j j alfbet)) ;temp=2*j+alfbet;
(setq a (* 2 j (+ j alfbet) (- temp 2.0))) ;a=2*j*(j+alfbet)*(temp-2.0);
(setq b (* (1- temp) (- (* alf alf) (* bet bet) (* temp (- 2 temp) z))));b=(temp-1.0)*(alf*alf-bet*bet+temp*(temp-2.0)*z);
(setq c (* 2 (+ j -1 alf) (+ j -1 bet) temp)) ;c=2.0*(j-1+alf)*(j-1+bet)*temp;
(setq p1 (/ (- (* b p2) (* c p3)) a)) ;p1=(b*p2-c*p3)/a;
(setq j (1+ j))
)
(setq pp (/ (+ (* N (- ALF BET (* TEMP Z)) P1)(* 2 (+ N ALF)(+ N BET) P2))
(* TEMP (- 1.0 (* Z Z)))
)
) ;pp=(n*(alf-bet-temp*z)*p1+2.0*(n+alf)*(n+bet)*p2)/(temp*(1.0-z*z));
(setq z1 z) ;z1=z;
(setq z (- z1 (/ p1 pp))) ;z=z1-p1/pp;
(setq its (1+ its))
(if (equal z z1 eps)
(setq flg nil)
)
)
(if (> its MAXIT)
(princ "\nToo many iterations in gaujac!") ;if (its > MAXIT) nrerror("too many iterations in gaujac");
)
(setq xi (cons (cons i z) xi)) ;x[i]=z;
(setq fi (exp (- (Math::gammln (+ alf n))
(- (Math::gammln (+ bet n)))
(Math::gammln (1+ n))
(Math::gammln (+ n alfbet 1))
)
)
)
(setq fi (/ (* fi temp (expt 2.0 alfbet)) pp p2)) ;w[i]=exp(gammln(alf+n)+gammln(bet+n)-gammln(n+1.0)-gammln(n+alfbet+1.0))*temp*pow(2.0,alfbet)/(pp*p2);
(setq wi (cons (cons i fi) wi))
(setq i (1+ i))
)
(MATH::INT:Bind xi wi)
)四、高斯-拉盖尔积分
功能: 计算广义积分 \[ \int_{0}^{\infty}e^{-x}f(x)\,dx\]
五、高斯-切比雪夫积分
此方法用于针对 \[\sqrt{(1-x^2)f(x)}\]型的积分有很高的效率
下面是一般函数的积分方法：
六 、龙贝格积分法：
七 辛普森积分法。
八、变步长梯形求积分法
九、步长积分法。
十、自适应积分法
一些相关函数：
下面是用对话框创建的求积分法的例程：
;;;=============================================================
;;; 用各种方法求积分的程序
;;;=============================================================
(defun C:Quadrature (/ ID OK DCL_FILE)
(setq id (load_dialog (setq Dcl_File (MATH::INT:Write_Dcl)))) ;从对话框中得到表达式
(vl-file-delete Dcl_File) ;删除临时对话框文件
(setq ok 2)
(if (new_dialog "dcl_Integration" id)
(progn
(VL-CATCH-ALL-APPLY 'MATH::INT:GetSettings) ;读取默认数据
(action_tile "help" "(MATH::INT:Help 1)") ;帮助
(foreach k '(0 1 2 3 4 5 6 7 8)
(setq k (strcat "K" (itoa k)))
(action_tile k "(MATH::INT:OnBtn $key)") ;按钮动作，对应相对的积分方法
)
(setq ok (start_dialog))
)
)
(unload_dialog ID)
(princ)
)
(defun C:JF (/)
(VL-CATCH-ALL-APPLY 'C:Quadrature)
(princ)
)
;;;=============================================================
;;; 从环境变量读取上次数据
;;;=============================================================
(defun MATH::INT:GetSettings (/ data)
(if (setq Data (getenv "Intergration"))
(foreach k (read data)
(set_tile (car k) (cdr k))
)
)
)
;;;=============================================================
;;; 检查对话框输入
;;;=============================================================
(defun MATH::INT:CheckInput (symS symA symB symN / e f)
(setq e (exp 1))
(set symS (get_tile "F"))
(set symA (MATH::INT:MyRead (get_tile "A")))
(set symB (MATH::INT:MyRead (get_tile "B")))
(set symN (MATH::INT:MyRead (get_tile "N")))
(setq f (CAL:Expr2Func (eval s) 'MATH::INT:func '(x)))
(apply 'and (mapcar 'eval '(F symS symA symB symN)))
)
;;;=============================================================
;;; 按钮动作，对应相应的函数求积分
;;;=============================================================
(defun MATH::INT:OnBtn (key / DATA s N a b OldZIN m EPS RET tm0 map msg foo tmp idx)
(setq m (VL-CATCH-ALL-APPLY 'MATH::INT:CheckInput '(s a b n)))
(if (or (vl-catch-all-error-p m) (not m) (equal a b 1e-8))
(if (vl-catch-all-error-p m)
(set_tile "info" (vl-catch-all-error-message m))
(set_tile "info" "无效输入!")
)
(progn
;;如果精度过高，设置为15位的精度
(if (> n 20)
(setq n 15)
(setq n (fix (abs n)))
)
;;如果上区间小于下区间，则交换区间
(if (< b a)
(setq tmp a a b b tmp)
)
;;记住对话框输入，用于下次
(setq OldZIN (getvar "DIMZIN"))
(setvar "DIMZIN" 8)
(set_tile "N" (itoa n))
(set_tile "A" (rtos a 2 20))
(set_tile "B" (rtos b 2 20))
(setq data (list (cons "F" s)
(cons "N" (itoa n))
(cons "A" (rtos a 2 20))
(cons "B" (rtos b 2 20))
)
)
(setvar "DIMZIN" OldZIN)
(setenv "Intergration" (VL-PRIN1-TO-STRING data))
;;开始计算积分
(setq eps (expt 0.1 n))
(setq tm0 (getvar "TDUSRTIMER"))
(setq map (MATH::INT:GetMethods)) ;积分计算方法集
(setq idx (atoi (substr key 2)))
(setq foo (nth idx map)) ;获取计算积分的函数
(setq ret (VL-CATCH-ALL-APPLY foo (list a b eps))) ;获取积分值
(if (vl-catch-all-error-p ret)
(set_tile "info" (vl-catch-all-error-message ret)) ;求解过程发生了错误
(if (null ret)
(set_tile "info" "发生了错误，求值结果为空!")
(progn
;(MATH::INT:Bench 100 a b eps)
(setq ret (rtos ret 2 20))
(setq msg (get_attr key "label"))
(setq msg (strcat msg "求的结果为:" ret))
(set_tile (strcat "R" (itoa idx)) ret) ;显示求解结果
(set_tile "info" msg) ;显示求解结果
(princ (strcat "\n" msg)) ;打印求解结果
(princ "\n费时:")
(princ (* (- (getvar "TDUSRTIMER") tm0) 86400))
(princ "秒.")
)
)
)
)
)
)
;;;=============================================================
;;; 各种积分测速
;;;=============================================================
(defun MATH::INT:Bench (n a b eps)
(UTI:BENCH
n
(list
(list 'MATH::INT:Romberg a b eps)
(list 'MATH::INT:Gauss-Legendre a b eps)
(list 'MATH::INT:Simpson a b eps)
)
)
)
;;;=============================================================
;;; 积分方法集
;;;=============================================================
(defun MATH::INT:GetMethods ()
'(MATH::INT:Romberg
MATH::INT:Simpson
MATH::INT:Atrapezia
MATH::INT:Trapezia
MATH::INT:Gauss-Legendre
Math::INT:Gauss-Chebyshev
Math::INT:Gauss-Laguerre
Math::INT:Gauss-Hermite
MATH::INT:Gauss-Jacobi
)
)
;;;=============================================================
;;; 表达式求值，也可以用cal函数
;;;=============================================================
(defun MATH::INT:MyRead (str / e)
(setq e (exp 1))
(CAL:Expr2Value str)
)
;;;=============================================================
;;; 帮助和说明: help and instruction
;;;=============================================================
(defun MATH::INT:Help (n)
(if (= n 1)
(if (= "CHS" (getvar "Locale"))
(alert
"函数式只接受符号x为变量,不规范很可能出错!
\n函数可以LISP内置的数学函数，也可以自定义函数!
\n指数用^表示，+-*/表示加减乘除，乘号不能省略。
\n程序能采用多种方法求积,一般来说龙贝格积分法最快。
\n有什么问题email: highflybird@qq.com
\n作者: highflybird 日期2019.07"
)
(alert
"Standard expression only accepts \"x\" as a variale!
\nThe fastest is Romberg Integration,the slowest is Trapezoidal rule(Be careful!).
\nRecommendation:Don't set a high precision at first,promote it step by step.
\nEspically for the Trapezoidal rule, It won't work well on some circumstances.
\nIt's an Open Source Software. Thanks for your advice or bug reports.
\nAuthor: highflybird Email: highflybird@qq.com Date:2019.07."
)
)
(set_tile "info" "表达式非法或者空输入.")
)
)对话框的制作：
;;;=============================================================
;;; 输入对话框
;;;=============================================================
(defun UTI:Inputbox (/ str wcs ret)
(setq str "Function GetNumbers()
GetNumbers=inputbox(\"请输入两个参数,中间用空格隔开:\",\"输入框\")
End Function"
)
(if
(or
(setq wcs (vlax-create-object "Aec32BitAppServer.AecScriptControl.1"))
(setq wcs (vlax-create-object "ScriptControl"))
)
(progn
(vlax-put-property wcs "language" "VBScript")
(vlax-invoke wcs 'addcode str)
(if (setq ret (vlax-invoke wcs 'run "GetNumbers"))
(setq ret (strcat "(" ret ")")
ret (read ret)
)
)
(vlax-release-object wcs)
(if
(and
(= 2 (length ret))
(or (= 'INT (type (car ret))) (= 'REAL (type (car ret))))
(or (= 'INT (type (cadr ret))) (= 'REAL (type (cadr ret))))
)
ret
)
)
)
)
;;;=============================================================
;;; 写对话框到文件用于程序
;;;=============================================================
(defun MATH::INT:Write_Dcl (/ Dcl_File file str)
(setq Dcl_File (vl-filename-mktemp nil nil ".Dcl"))
(setq file (open Dcl_File "w"))
(princ
"dcl_Integration : dialog {
label = \"数值积分LISP版 v1.2\";
: boxed_column {
width = 60;
fixed_width = true;
: edit_box {
key=\"F\";
label= \"函数:\";
}
: row {
: edit_box {
key=\"A\";
label= \"下限:\";
}
: edit_box {
key=\"B\";
label= \"上限:\";
}
: edit_box {
key=\"N\";
label = \"精确位数:\";
value = 8;
edit_width = 2;
fixed_width = true;
}
}
spacer_1;
}
: row {
: boxed_column {
label = \"计算方法:\";
: button {
key = \"K0\";
label = \"龙贝格积分法\";
}
: button {
key = \"K1\";
label = \"辛普森积分法\";
}
: button {
key = \"K2\";
label = \"自适应积分法\";
}
: button {
key = \"K3\";
label = \"变步长梯形法\";
}
: button {
key = \"K4\";
label = \"高斯-勒让德法\";
}
: button {
key = \"K5\";
label = \"高斯-切比雪夫法\";
}
: button {
key = \"K6\";
label = \"高斯-拉盖尔法\";
}
: button {
key = \"K7\";
label = \"高斯-埃尔米特法\";
}
: button {
key = \"K8\";
label = \"高斯-雅克比法\";
}
spacer;
}
: boxed_column {
width = 32;
fixed_width = true;
label = \"计算结果:\";
: text {
key = \"R0\";
}
: text {
key = \"R1\";
}
: text {
key = \"R2\";
}
: text {
key = \"R3\";
}
: text {
key = \"R4\";
}
: text {
key = \"R5\";
}
: text {
key = \"R6\";
}
: text {
key = \"R7\";
}
: text {
key = \"R8\";
}
spacer;
}
}
ok_cancel_help;
//ok_cancel_help_errtile;
: text {
key = \"info\";
label = \"Copyright \\u+00A9 2007-2019 Highflybird. All rights reserved.\";
width = 20;
}
} "
file
)
(close file)
Dcl_File
)
;;;=============================================================
;;; 下面的就不用介绍了
;;;=============================================================
(vl-load-com)
(if (= "CHS" (getvar "Locale"))
(prompt "输入命令: JF")
(prompt "Please enter: Quadrature")
)
(c:JF)
(princ)一些应用，如下面网友的提问并解答：
http://bbs.mjtd.com/thread-179809-1-1.html
椭球体球缺的表面积计算
用积分公式用LISP程序表达并计算出来
这是一个网上的
有结果和公式
图像
用上面的介绍的一些函数可以得出其面积和体积：
;;;=============================================================
;;; 功能: 主程序，获取椭球冠的面积
;;; 输入: 无。
;;; 输出: 无。
;;;=============================================================
(defun c:test (/ a b c d h s0 s1)
(initget 15)
(setq a (getdist "\n输入椭球X半轴长: "))
(initget 15)
(setq b (getdist "\n输入椭球y半轴长: "))
(initget 15)
(setq c (getdist "\n输入椭球z半轴长: "))
(initget 15)
(setq d (getdist "\n输入椭球冠的高度: "))
(if (<= d (+ c c)) ;所求范围为小于椭球Z轴长
(progn
(setq h (/ (- c d) c))
(setq s0 (ELL:GetSurfaceArea a b c 1)) ;用龙贝格积分法计算其半个椭球面积
(setq s2 (ELL:GetSurfaceArea a b c 1)) ;用高斯-切比雪夫积分法计算其半个椭球面积
(if (zerop h) ;为了防止被零除
(setq s1 0 s2 0)
(setq s1 (ELL:GetSurfaceArea a b c h) ;用龙贝格积分法计算椭球带面积
s3 (ELL:GetSurfaceArea3 a b c h) ;用高斯-切比雪夫积分法计算椭球带面积
)
)
(princ "\n用龙贝格法得到所求表面积是: ")
(princ (rtos (- s0 s1) 2 20)) ;两个面积相减，便是其椭球冠面积
(princ "\n用高斯-切比雪夫积分法得到所求表面积是: ")
(princ (rtos (- s2 s3) 2 20)) ;两个面积相减，便是其椭球冠面积
(princ "\n用龙贝格法所求的椭球的表面积是: ")
(princ (rtos (+ s0 s0) 2 20))
(princ "\n估算结果是: ")
(princ (rtos (ELL:GetSurfaceArea2 a b c) 2 20))
(princ "\n另外的结果是: ")
(princ (rtos (ELL:GetSurfaceArea1 a b c) 2 20))
(princ "\n高斯-切比雪夫积分的结果是: ")
(princ (rtos (+ s2 s2) 2 20))
)
)
(princ)
)
;;;=============================================================
;;; 功能: 获取椭球带的面积
;;; 输入: 椭球的三个方向的轴径，和椭球带的高度
;;; 输出: 椭球带的面积
;;;=============================================================
(defun ELL:GetSurfaceArea (a b c h / func)
(defun Func (x a b c h / F G K S Y Z)
(setq k (* a a))
(setq s (sin x))
(setq g (- 1 (* (- 1 (/ k b b)) s s)))
(setq z (1- (/ k c c g)))
(setq y (1+ (* z h h)))
(if (zerop z)
(* (sqrt g) (1+ (sqrt y)))
(if (< z 0)
(setq z (* h (sqrt (- z)))
f (* (sqrt g) (+ (sqrt y) (/ (asin z) z)))
)
(setq z (* h (sqrt z))
f (* (sqrt g) (+ (sqrt y) (/ (asinh z) z)))
)
)
)
)
(* 2 b c h (Math:Romberg 'func (list a b c h) 0 (* 0.5 pi) 1e-15))
)
;;;=============================================================
;;; 功能: 用椭圆函数计算椭球表面积
;;;=============================================================
(defun ELL:GetSurfaceArea1 (a b c / ll aa bb cc ac e1 e2 ph k f1 f2 s)
(if (and (equal a b 1e-8) (equal b c 1e-8))
(* 4 pi a)
(setq ll (vl-sort (list a b c) '>)
a (car ll)
b (cadr ll)
c (caddr ll)
aa (* a a)
bb (* b b)
cc (* c c)
ac (sqrt (- aa cc))
e1 (/ ac a)
e2 (/ (sqrt (- bb cc)) b)
ph (asin e1)
k (/ e2 e1)
f1 (Math:Elliptic_Integral_1 ph k)
f2 (Math:Elliptic_Integral_2 ph k)
s (* 2 pi (+ cc (/ (* b cc f1) ac) (* b ac f2)))
)
)
)
;;;=============================================================
;;; 功能: 估算椭球表面积
;;;=============================================================
(defun ELL:GetSurfaceArea2 (a b c)
(* 4
pi
(expt
(/ (+ (expt (* a b) 1.6075)
(expt (* b c) 1.6075)
(expt (* c a) 1.6075)
)
3
)
(/ 1 1.6075)
)
)
)
;;;=============================================================
;;; 几个相关数学函数
;;;=============================================================
(defun asin (x) (atan x (sqrt (- 1 (* x x))))) ;反正弦函数
(defun asinh (x) (log (+ x (sqrt (1+ (* x x)))))) ;反双曲正弦函数=log(x+sqrt(x*x+1))
(vl-load-com)
(princ "\n运行命令是:Test")
(princ)至于一些积分函数有什么区别，我就不在这里一一介绍了。
希望这些代码对你有帮助。引用请注明来源。