{"id":4815,"date":"2020-02-29T12:20:55","date_gmt":"2020-02-29T04:20:55","guid":{"rendered":"https:\/\/www.highflybird.com\/blog\/?p=4815"},"modified":"2020-09-16T21:35:55","modified_gmt":"2020-09-16T13:35:55","slug":"%e6%95%b0%e5%80%bc%e8%ae%a1%e7%ae%97%e4%b9%8b%e6%b1%82%e7%a7%af%e5%88%86","status":"publish","type":"post","link":"https:\/\/www.highflybird.com\/blog\/?p=4815","title":{"rendered":"\u6570\u503c\u8ba1\u7b97\u4e4b\u6c42\u79ef\u5206"},"content":{"rendered":"\n

\u7528LISP\u7f16\u5199\u4e86\u4e00\u4e2a\u6c42\u79ef\u5206\u7684\u7a0b\u5e8f\uff1a<\/p>\n\n\n\n

\u91cc\u9762\u91c7\u7528\u4e86\u5404\u79cd\u65b9\u6cd5\u6c42\u79ef\u5206\u548c\u5404\u79cd\u7c7b\u578b\u7684\u79ef\u5206\u3002\u4e0b\u9762\u6211\u628a\u5404\u79cd\u65b9\u6cd5\u7684\u6e90\u7801\u8d34\u51fa\u3002 <\/p>\n\n\n

<\/p>\n\n\n

\u65b9\u6cd5\u4e00\uff1a \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u79ef\u5206\u6cd5\u3002<\/p>\n\n\n

[codesyntax lang=”lisp”]<\/p>\n

;;;=============================================================\n;;; \u8ba1\u7b97\u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570\u7684\u7cfb\u6570 \uff0c\u5982\u4e0b\u9762\u76846\u6b21\u9879\u7cfb\u6570           \n;;; p0(x) = 1                                                   \n;;; p1(x) = x                                                   \n;;; p2(x) = (3*x^2-1)\/2                                         \n;;; p3(x) = (5*x^3-3*x)\/2                                       \n;;; p4(x) = (35*x^4-30*x^2+3)\/8                                 \n;;; p5(x) = (63*x^5-70*x^3+15*x)\/8                              \n;;; p6(x) = (231*x^6-315*x^4+105*x^2-5)\/16                      \n;;; x-2                                                         \n;;; x-1                                                         \n;;; 9x-5                                                        \n;;; 216*x^2-216*x+49                                            \n;;; 45000*x^2-32200*x+5103                                      \n;;; 2025000*x^3-2025000*x^2+629325*x-58564                      \n;;; 142943535000*x^3-1130712534000*x^2+27510743799*x-1976763932 \n;;; \u8f93\u5165: x1,x2 \u533a\u95f4(\u4e00\u822c\u6765\u8bf4\u662f-1..1)\uff0cn \u8fed\u4ee3\u6b21\u6570,eps\u8fed\u4ee3\u7cbe\u5ea6   \n;;; \u8f93\u51fa: \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570\u7684\u7cfb\u6570,\u7528\u70b9\u8868\u96c6\u8868\u793a                \n;;; http:\/\/mathworld.wolfram.com\/Legendre-GaussQuadrature.html  \n;;;=============================================================\n(defun Math::Int:Legendre_Polynomial (x1 x2 n eps \/ xi wi FI I ITER J M P1 P2 P3 PP XL XM Z Z1)\n  (setq m (\/ (1+ n) 2))\n  (setq xm (* 0.5 (+ x2 x1)))\n  (setq xl (* 0.5 (- x2 x1)))\n  (setq i 1)\n  (repeat m\n    (setq z (cos (\/ (* pi (- i 0.25)) (+ n 0.5))))\n    (setq iter 0)\n    (while (and (not (equal z z1 eps)) (< iter 1000))\n      (setq p1 1.0)\n      (setq p2 0.0)\n      (setq j 1)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (\/ (- (* z p2 (+ j j -1.)) (* (1- j) p3)) j))\n\t(setq j (1+ j))\n      )\n      (setq pp (\/ (* n (- (* z p1) p2)) (1- (* z z))))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq iter (1+ iter))\n    )\n    (setq fi (\/ xl 0.5 (- 1 (* z z)) pp pp))\n    (setq xi (cons (cons (1- i) (- xm (* xl z))) xi))\n    (setq wi (cons (cons (1- i) fi) wi))\n    (if (\/= (1- i) (- n i))\n      (setq xi (cons (cons (- n i) (+ xm (* xl z))) xi)\n\t    wi (cons (cons (- n i) fi) wi)\n      )\n    )\n    (setq i (1+ i)) \n  )\n  (MATH::INT:Bind xi wi)\n)\n\n;;;=============================================================\n;;; \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570                                         \n;;;=============================================================\n(defun MATH::INT:Gauss-Legendre (a b eps \/ AA BB EP FX G H I L M P S W X)\n  (setq l '((-0.93246951420315202787 . 0.17132449237917034504)  \n\t    (-0.66120938646626451363 . 0.36076157304813860756)     \n\t    (-0.23861918608319690859 . 0.46791393457269104739)  \n\t    ( 0.23861918608319690859 . 0.46791393457269104739)     \n            ( 0.66120938646626451363 . 0.36076157304813860756)     \n\t    ( 0.93246951420315202787 . 0.17132449237917034504)\n\t  )\n  )                                                         \n  ;(setq L (Math::Int:Legendre_Polynomial -1 1 100 2e-20))\n  (setq m 1)\n  (setq h (- b a))\n  (setq s (abs (* 0.001 h)))\n  (setq p 1e100) \t\n  (setq ep (1+ eps))\n  (while (and (>= ep eps) (> (abs h) s))\n    (setq g 0)\n    (setq i 1)\n    (repeat m\n      (setq bb (+ a (* i h)))\n      (setq aa (- bb h))\n      (setq w 0)\n      (foreach k l\n\t(setq x (* 0.5 (+ bb aa (* (- bb aa) (car k)))))\n\t(setq fx (MATH::INT:func x))\n\t(setq w (+ w (* fx (cdr k))))\n      )\n      (setq g (+ g w))\n      (setq i (1+ i))\n    ) \n    (setq g (* g h 0.5))\n    (setq ep (\/ (abs (- g p)) (1+ (abs g))))\n    (setq p g)\n    (setq m (1+ m))\n    (setq h (\/ (- b a) m))\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570(\u53e6\u4e00\u65b9\u6cd5\uff0c\u6162\u4e9b)                         \n;;;=============================================================\n(defun MATH::INT:Gauss-Legendre1 (a b eps \/ FLAG G H L N X Y)\n  (setq n 1)                                                      \n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (while (and (< n 100) flag)\t\n    (setq g 0)\n    (setq L (Math::Int:Legendre_Polynomial a b n eps))\n    (foreach w L\n      (setq x (car w))\n      (setq y (MATH::INT:func x))\n      (setq g (+ g (* (cdr w) y)))\n    )\n    (if (equal g h eps)\n      (setq flag nil)\n      (setq n (+ n n)\n\t    h g\n      )\n    )\n  )\n  g\n)[\/codesyntax]<\/pre>\n\n\n
\u65b9\u6cd5\u4e8c\uff1a\u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206<\/p>\n\n\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206                                           \n;;; \u529f\u80fd: \u8ba1\u7b97 e^(-x^2)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f4-INF..INF\u4e0a)       \n;;;=============================================================\n(defun Math::INT:Gauss-Hermite (a b eps \/ n L g x y)\n  (setq n 100)\n  (setq L (Math::INT:GetHermite n))\n  (setq g 0)\n  (foreach w L\n    (setq x (car w))\n    (setq y (MATH::INT:func x))\n    (setq g (+ g (* (cdr w) y)))\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u83b7\u53d6\u57c3\u7c73\u5c14\u7279\u7cfb\u6570                                            \n;;;=============================================================\n(defun Math::INT:GetHermite (n \/ xi wi EPS FI I ITS J M MAXIT P1 P2 P3 PIM4 PP Z Z1)\n  (setq eps 1e-15)\n  (setq PIM4 0.7511255444649425)\n  (setq maxIt 10)\n  (setq m (\/ (1+ n) 2))\n  (setq i 0)\n  (while (< i m)\n    (if (= i 0)\n      (setq z (- (sqrt (+ n n 1)) (* 1.85575 (expt (+ n n 1) (\/ -1 6.)))))\n      (if (= i 1)\n\t(setq z (- z (\/ (* 1.14 (expt n 0.426)) z)))\n\t(if (= i 2)\n\t  (setq z (- (* 1.86 z) (* 0.86 (cdr (assoc 0 xi)))))\n\t  (if (= i 3)\n\t    (setq z (- (* 1.91 z) (* 0.91 (cdr (assoc 1 xi)))))\n\t    (setq z (- (+ z z) (cdr (assoc (- i 2) xi))))\n\t  )\n\t)\n      )\n    )\n    (setq its 0)\n    (while (and (not (equal z z1 eps)) (< its MAXIT))\n      (setq p1 pIM4)\n      (setq p2 0.0)\n      (setq j 0)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (- (* z p2 (sqrt (\/ 2.0 (1+ j)))) (* p3 (sqrt (\/ j (+ 1.0 j))))))\n\t(setq j (1+ j))\n      )\n      (setq pp (* p2 (sqrt (+ n n))))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq its (1+ its))\n    )\n    (setq fi (\/ 2.0  pp pp))\n    (setq xi (cons (cons i z) xi))\n    (setq wi (cons (cons i fi) wi))\n    (if (\/= i (- n 1 i))\n      (setq xi (cons (cons (- n 1 i) (- z)) xi)\n\t    wi (cons (cons (- n 1 i) fi) wi)\n      )\n    )\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)[\/codesyntax]
<\/pre>\n\n\n
\u65b9\u6cd5\u4e09\uff1a\u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206<\/p>\n\n\n\n
\u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206<\/p>\n\n\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97 f(x)*((1-x)^a)*((1+x)^b)\u7684\u79ef\u5206(\u5728\u533a\u95f4-1..1\u4e0a)    \n;;;=============================================================\n(defun MATH::INT:Gauss-Jacobi (a b eps \/ ALF ARGS BET G N X Y flag g0)\n  (if (setq args (UTI:InputBox))\n    (progn\n      (setq flag T)\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n      (setq n 10)\n      (while (and (< n 100) flag)\t\n        (setq alf (car args))\n        (setq bet (cadr args))\n        (setq g 0)\n        (foreach w (MATH::INT:GetJacobiPolynomial n alf bet)\n          (setq x (car w))\n          (setq y (MATH::INT:func x))\n          (setq g (+ g (* (cdr w) y)))\n        )\n\t(if (equal g g0 eps)\n\t  (setq flag nil g0 g)\n\t  (setq n (+ n n) g0 g)\n\t)\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97\u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206\u7cfb\u6570\u9879                             \n;;;=============================================================\n(defun MATH::INT:GetJacobiPolynomial (n alf bet \/\n\t\t\t\t      xi wi A ALFBET AN B BN C EPS FI FLG I ITS\n\t\t\t\t      J MAXIT P1 P2 P3 PP R1 R2 R3 TEMP Z Z1)\n  (setq maxIT 40)\n  (setq eps 1e-15)\n  (setq alf (float alf))\n  (setq bet (float bet))\n  (setq i 0)\n  (repeat n\t\t\t\t\t\t\t\t\t;for (i=0;i<n;i++) {\n    (cond\n      ( (= i 0)\t\t\t\t\t\t\t\t\t;if (i == 0) {\n        (setq an (\/ alf n))\t\t\t\t\t\t\t;an=alf\/n;\n        (setq bn (\/ bet n))\t\t\t\t\t\t\t;bn=bet\/n;\n        (setq r1 (* (1+ alf) (+ (\/ 2.78 (+ 4.0 (* n n))) (\/ (* 0.768 an) n))))\t;r1=(1.0+alf)*(2.78\/(4.0+n*n)+0.768*an\/n);\n        (setq r2 (+ 1.0 (* 1.48 n) (* 0.96 bn) (* 0.452 an an) (* 0.83 an bn)))\t;r2=1.0+1.48*an+0.96*bn+0.452*an*an+0.83*an*bn;\n        (setq z  (- 1 (\/ r1 r2)))\t\t\t\t\t\t;z=1.0-r1\/r2;\n      )\n      ( (= i 1)\t\t\t\t\t\t\t\t\t;else if (i == 1)\n        (setq r1 (\/ (+ 4.1 alf) (* (1+ alf) (1+ (* 0.156 alf)))))\t\t;r1=(4.1+alf)\/((1.0+alf)*(1.0+0.156*alf));\n        (setq r2 (1+ (\/ (* 0.06 (- n 8.0) (1+ (* 0.12 alf))) n)))\t\t;r2=1.0+0.06*(n-8.0)*(1.0+0.12*alf)\/n;\n        (setq r3 (1+ (\/ (* 0.012 bet (1+ (* 0.25 (abs alf)))) n)))\t\t;r3=1.0+0.012*bet*(1.0+0.25*fabs(alf))\/n;\n        (setq z  (- z (* (- 1 z) r1 r2 r3)))\t\t\t\t\t;z -= (1.0-z)*r1*r2*r3;\n      )\n      ( (= i 2)\t\t\t\t\t\t\t\t\t;else if (i == 2)\n        (setq r1 (\/ (+ 1.67 (* 0.28 alf)) (1+ (* 0.37 alf))))\t\t\t;r1=(1.67+0.28*alf)\/(1.0+0.37*alf);\n        (setq r2 (1+ (\/ (* 0.22 (- n 8.0)) n)))\t\t\t\t\t;r2=1.0+0.22*(n-8.0)\/n;\n        (setq r3 (1+ (\/ (* 8.0 bet) (* (+ 6.28 bet) n n))))\t\t\t;r3=1.0+8.0*bet\/((6.28+bet)*n*n);\n        (setq z  (- z (* (- (cdr (assoc 0 xi)) z) r1 r2 r3)))\t\t\t;z -= (x[0]-z)*r1*r2*r3;\n      )\n      ( (= i (- n 2))\t\t\t\t\t\t\t\t;else if (i == n-2)\n        (setq r1 (\/ (1+ (* 0.235 bet)) (+ 0.766 (* 0.119 bet))))\t\t;r1=(1.0+0.37*bet)\/(1.67+0.28*bet);\n        (setq r2 (\/ 1.0 (1+ (\/ (* 0.639 (- n 4.0)) (1+ (* 0.71 (- n 4.0)))))))\t;r2=1.0\/(1.0+0.639*(n-4.0)\/(1.0+0.71*(n-4.0)));\n        (setq r3 (\/ 1.0 (1+ (\/ (* 20.0 alf) (* (+ 7.5 alf) n n))))) \t\t;r3=1.0\/(1.0+20.0*alf\/((7.5+alf)*n*n));\n        (setq z  (+ z (* (- z (cdr (assoc (- n 4) xi))) r1 r2 r3))) \t\t;z += (z-x[n-4])*r1*r2*r3;\n      )\n      ( (= i (1- n))\t\t\t\t\t\t\t\t;else if (i == n-1)\n        (setq r1 (\/ (1+ (* 0.37 bet))(+ 1.67 (* 0.28 bet))))\t\t\t;r1=(1.0+0.37*bet)\/(1.67+0.28*bet);\n        (setq r2 (\/ 1.0 (1+ (\/ (* 0.22 (- n 8.0)) n))))\t\t\t\t;r2=1.0\/(1.0+0.22*(n-8.0)\/n);\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));\n       \t(setq z  (+ z (* (- z (cdr (assoc (- n 3) xi))) r1 r2 r3))) \t\t;z += (z-x[n-3])*r1*r2*r3;\n      )\n      (t\t\t\t\t\t\t\t\t\t;else {\n        (setq z (* 3 (- (cdr (assoc (1- i) xi)) (cdr (assoc (- i 2) xi)))))\t;z=3.0*x[i-1]-3.0*x[i-2]+x[i-3];\n        (setq z (+ z (cdr (assoc (- i 3) xi))))\n      )\n    )\n    (setq alfbet (+ alf bet))\t\t\t\t\t\t\t;alfbet=alf+bet;\n    (setq its 1)\n    (setq flg T)\n    (while (and flg (<= its maxIt))\t\t\t\t\t\t;for (its=1;its<=MAXIT;its++)\n      (setq temp (+ 2.0 alfbet))\t\t\t\t\t\t;temp=2.0+alfbet;\n      (setq p1 (\/ (+ (- alf bet) (* temp z)) 2.0))\t\t\t\t;p1=(alf-bet+temp*z)\/2.0;\n      (setq p2 1.0)\t\t\t\t\t\t\t\t;p2=1.0;\n      (setq j 2)\t\t\t\t\t\t\t\t\n      (while (<= j n)\t\t\t\t\t\t\t\t;for (j=2;j<=n;j++) {\n\t(setq p3 p2)\t\t\t\t\t\t\t\t;p3=p2;\n\t(setq p2 p1)\t\t\t\t\t\t\t\t;p2=p1;\n\t(setq temp (+ j j alfbet))\t\t\t\t\t\t;temp=2*j+alfbet;\n\t(setq a (* 2 j (+ j alfbet) (- temp 2.0)))\t\t\t\t;a=2*j*(j+alfbet)*(temp-2.0);\n\t(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);\n\t(setq c (* 2 (+ j -1 alf) (+ j -1 bet) temp))\t\t\t\t;c=2.0*(j-1+alf)*(j-1+bet)*temp;\n        (setq p1 (\/ (- (* b p2) (* c p3)) a))\t\t\t\t\t;p1=(b*p2-c*p3)\/a;\n\t(setq j (1+ j))\n      )\n      \n      (setq pp (\/ (+ (* N (- ALF BET (* TEMP Z)) P1)(* 2 (+ N ALF)(+ N BET) P2))\n\t\t  (* TEMP (- 1.0 (* Z Z)))\n\t       )\n      )\t\t\t\t\t\t\t\t\t\t;pp=(n*(alf-bet-temp*z)*p1+2.0*(n+alf)*(n+bet)*p2)\/(temp*(1.0-z*z));\n      (setq z1 z)\t\t\t\t\t\t\t\t;z1=z;\n      (setq z (- z1 (\/ p1 pp)))\t\t\t\t\t\t\t;z=z1-p1\/pp;\n      (setq its (1+ its))\n      (if (equal z z1 eps)\n\t(setq flg nil)\n      )\n    )\n    (if (> its MAXIT)\n      (princ \"\\nToo many iterations in gaujac!\")\t\t\t\t;if (its > MAXIT) nrerror(\"too many iterations in gaujac\");\n    )\n    (setq xi (cons (cons i z) xi))\t\t\t\t\t\t;x[i]=z;\n    (setq fi (exp (- (Math::gammln (+ alf n))\n\t\t     (- (Math::gammln (+ bet n)))\n\t\t     (Math::gammln (1+ n))\n\t\t     (Math::gammln (+ n alfbet 1))\n\t\t  )\n\t     )\n    )\n    (setq fi (\/ (* fi temp (expt 2.0 alfbet)) pp p2))\t\t\t\t;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);\t\n    (setq wi (cons (cons i fi) wi))\t\t\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)<\/pre>\n
[\/codesyntax]<\/p>\n
\u56db\u3001\u9ad8\u65af-\u62c9\u76d6\u5c14\u79ef\u5206 
\u529f\u80fd: \u8ba1\u7b97 e^(-x)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f40..INF\u4e0a)<\/p>\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u62c9\u76d6\u5c14\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97 e^(-x)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f40..INF\u4e0a)            \n;;;=============================================================\n(defun Math::INT:Gauss-Laguerre (a b eps \/ n flag L g h x y maxN)\n  (setq n 2)\n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (setq maxN 40)\n  (while (and flag (< n maxN))\t\n    (setq g 0)\n    (setq L (Math::INT:GetLaguerre n 0))\n    (foreach w L\n      (setq x (car w))\n      (setq y (MATH::INT:func x))\n      (setq g (+ g (* (cdr w) y)))\n    )\n    (if (equal g h eps)\n      (setq flag nil)\n      (setq n (+ n n)\n\t    h g\n      )\n    )\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u83b7\u53d6\u62c9\u76d6\u5c14\u7cfb\u6570                                              \n;;;=============================================================\n(defun Math::INT:GetLaguerre (n alf \/ xi wi AI EPS I ITS J MAXIT P1 P2 P3 PP X XI2 YI Z Z1)\n  (setq maxIt 10)\n  (setq eps 1e-15)\n  (setq i 0)\n  (repeat n\n    (if\t(= i 0)\n      (setq z (\/ (* (1+ alf) (+ 3 (* 0.92 alf)))\n\t\t (+ 1 (* 2.4 n) (* 1.8 alf))\n\t      )\n      )\n      (if (= i 1)\n\t(setq\n\t  z (+ z (\/ (+ 15 (* 6.25 alf)) (+ 1 (* 0.9 alf) (* 2.5 n))))\n\t)\n\t(setq ai  (1- i)\n\t      xi2 (cdr (assoc (- i 2) xi))\n\t      x\t  (\/ (*\n\t\t       (+\n\t\t\t (\/ (1+ (* 2.55 AI)) (* 1.9 AI))\n\t\t\t (\/ (* 1.26 AI ALF) (1+ (* 3.5 AI)))\n\t\t       )\n\t\t       (- Z XI2)\n\t\t     )\n\t\t     (1+ (* 0.3 ALF))\n\t\t  )\n\t      z\t  (+ z x)\n\t)\n      )\n    )\n    (setq its 0)\n    (while (and (not (equal z z1 eps)) (< its maxIt))\n      (setq p1 1)\n      (setq p2 0)\n      (setq j 0)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (\/ (- (* (+ J J 1.0 ALF (- Z)) P2) (* (+ J ALF) P3))\n\t\t    (1+ J)\n\t\t )\n\t)\n\t(setq j (1+ j))\n      )\n      (setq pp (\/ (- (* n p1) (* p2 (+ n alf))) z))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq its (1+ its))\n    )\n    (setq xi (cons (cons i z) xi))\n    (setq yi (-\t(\/ (exp (- (Math::gammln (+ alf n)) (Math::gammln n)))\n\t\t   (* pp n p2)\n\t\t)\n\t     )\n    )\n    (setq wi (cons (cons i yi) wi))\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)
[\/codesyntax]
\u4e94\u3001\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206 
\u6b64\u65b9\u6cd5\u7528\u4e8e\u9488\u5bf9 sqrt(1-x^2)*f(x)\u578b\u7684\u79ef\u5206\u6709\u5f88\u9ad8\u7684\u6548\u7387
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206                                           \n;;; \u6b64\u65b9\u6cd5\u7528\u4e8e\u9488\u5bf9 sqrt(1-x^2)*f(x)\u578b\u7684\u79ef\u5206\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002       \n;;; \u8f93\u5165: foo \u51fd\u6570\u540d,arg \u9664\u81ea\u53d8\u91cfx\u5916\u7684\u53c2\u6570\u5217\u8868, n\u8fed\u4ee3\u6b21\u6570\u3002     \n;;; \u8f93\u51fa: \u6b64\u7c7b\u79ef\u5206\u7684\u6570\u503c.                                       \n;;; \u8bf4\u660e: \u6b64\u79ef\u5206\u6cd5\u6548\u7387\u8f83\u9ad8\uff0cn\u53d6\u503c\u4e00\u822c10\u6b21\u5de6\u53f3\u5c31\u8fbe\u5230\u6709\u6548\u6d6e\u70b9\u7cbe\u5ea6.\n;;; http:\/\/mathworld.wolfram.com\/Chebyshev-GaussQuadrature.html \n;;;=============================================================\n(defun Math::INT:Gauss-Chebyshev (a b eps \/ FI FLAG I N S0 SX WI XI)\n  (setq n 3)\t\t\t\t\t\t\t;\u4e00\u822c\u6765\u8bf4\u8fed\u4ee36-5\u6b21\u5de6\u53f3\u5c31\u53ef\u4ee5\u8fbe\u5230\u6d6e\u70b9\u8ba1\u7b97\u7cbe\u5ea6\n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (while (and (< n 1000) flag)\t\t\t\t\t;\u8fed\u4ee3\u6b21\u6570\u4e0d\u8d85\u8fc7100\n    (setq wi (\/ pi n))\t\t\t\t\t\t;\u6240\u6709\u7684\u6743\u503c\u5747\u4e3api\/n\n    (setq sx 0)\n    (setq i 1)\n    (repeat n\n      (setq xi (cos (\/ (* pi (- i 0.5)) n)))\t\t\t;x \u9879\n      (setq fi (MATH::INT:func xi))\t\t\t\t;x \u9879\u7684\u51fd\u6570\u503c\n      (setq sx (+ sx fi))\n      (setq i  (1+ i))\n    )\n    (setq sx (* wi sx))\t\t\t\t\t\t;\u7edf\u4e00\u4e58\u4ee5\u6743\u503c\n    (if (equal sx s0 eps)\t\t\t\t\t;\u662f\u5426\u6ee1\u8db3\u7cbe\u5ea6\u8981\u6c42\n      (setq flag nil)\t\t\t\t\t\t;\u662f\u5219\u4e0d\u518d\u8fed\u4ee3\n      (setq n (+ n n)\t\t\t\t\t\t;\u5426\u5219\u8fed\u4ee3\u6b21\u6570\u500d\u589e\n\t    s0 sx\t\t\t\t\t\t;\u5b58\u50a8\u79ef\u5206\u503c\n      )\n    )\n  )\n  sx\n)<\/pre>\n
[\/codesyntax]

\u4e0b\u9762\u662f\u4e00\u822c\u51fd\u6570\u7684\u79ef\u5206\u65b9\u6cd5\uff1a
\u516d \u3001\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u9f99\u8d1d\u683c\u79ef\u5206\u6cd5                                                \n;;; Romberg Integration                                         \n;;; \u8f93\u5165: \u51fd\u6570\u540d--foo (\u7528\u7b26\u53f7\u8868\u793a\uff0c\u4e00\u822c\u5f62\u5f0f\u662f (foo x a b c ...) \n;;;       \u53c2\u6570\u8868--args \uff0c\u9664\u53bb\u81ea\u53d8\u91cfx\u7684\u5176\u5b83\u7684\u53c2\u6570\u5217\u8868            \n;;;       \u4e0b\u533a\u95f4--Ra                                            \n;;;       \u4e0a\u533a\u95f4--Rb                                            \n;;;       \u5bb9\u8bb8\u8bef\u5dee--eps                                         \n;;; \u8f93\u51fa: \u6240\u6c42\u51fd\u6570\u5728\u533a\u95f4\u6bb5\u7684\u79ef\u5206                                \n;;;=============================================================\n(defun MATH::INT:Romberg (a b eps \/ EP H I K M N P Q S X Y Y0)\n  (setq h (- b a))\n  (setq y nil)\n  (setq i 0)\n  (repeat 20\n    (setq y (cons (cons i 0.0) y))\n    (setq i (1+ i))\n  )\n  (setq y (reverse y))\n  (setq y0 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq y (cons (cons 0 y0) (cdr y)))\n  (setq\tm  1\n\tn  1\n\tep (1+ eps)\n  )\n  (while (and (>= ep eps) (<= m 19))\n    (setq p 0.0)\n    (setq i 0)\n    (repeat n\n      (setq x (+ a (* (+ i 0.5) h)))\n      (setq p (+ p (MATH::INT:func x)))\n      (setq i (1+ i))\n    )\n    (setq p (\/ (+ (cdar y) (* h p)) 2.0))\n    (setq s 1.0)\n    (setq k 1)\n    (repeat m\n      (setq s (+ s s s s))\n      (setq q (\/ (- (* s p) (cdr (assoc (1- k) y))) (1- s)))\n      (setq y (subst (cons (1- k) p) (assoc (1- k) y) y))\n      (setq p q)\n      (setq k (1+ k))\n    )\n    (setq ep (abs (- q (cdr (assoc (1- m) y)))))\n    (setq m (1+ m))\n    (setq y (subst (cons (1- m) q) (assoc (1- m) y) y))\n    (setq n (+ n n))\n    (setq h (\/ h 2.0))\n  )\n  q\n)<\/pre>\n
[\/codesyntax]
\u4e03 \u8f9b\u666e\u68ee\u79ef\u5206\u6cd5\u3002
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u8f9b\u666e\u68ee\u79ef\u5206\u6cd5                                                \n;;; Simpson Integration                                         \n;;;=============================================================\n(defun MATH::INT:Simpson (a b eps \/ EP H ITER K N P S1 S2 T1 T2 X)\n  (setq n 1)\n  (setq h (- b a))\n  (setq t1 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq s1 t1)\n  (setq ep (1+ eps))\n  (setq iter 0)\n  (while (and (>= ep eps) (< iter 50))\n    (setq p 0.0)\n    (setq k 0)\n    (repeat n\n      (setq x (+ a (* (+ k 0.5) h)))\n      (setq p (+ p (MATH::INT:func x)))\n      (setq k (1+ k))\n    )\n    (setq t2 (\/ (+ t1 (* h p)) 2.))\n    (setq s2 (\/ (- (* 4.0 t2) t1) 3.))\n    (setq ep (abs (- s2 s1)))\n    (setq t1 t2)\n    (setq s1 s2)\n    (setq n (+ n n))\n    (setq h (\/ h 2))\n    (setq iter (1+ iter))\n  )\n  s2\n)\n<\/pre>\n
[\/codesyntax]
\u516b\u3001\u53d8\u6b65\u957f\u68af\u5f62\u6c42\u79ef\u5206\u6cd5 
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u53d8\u6b65\u957f\u68af\u5f62\u6c42\u79ef\u5206\u6cd5                                          \n;;; Trapezoidal Integration 1                                   \n;;;=============================================================\n(defun MATH::INT:Trapezia (a b eps \/ H K N P S T1 T2 X iter)\n  (setq n 1)\n  (setq h (- b a))\n  (setq t1 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq p (1+ eps))\n  (setq iter 0)\n  (while (and (>= p eps) (< iter 100))\n    (setq s 0)\n    (setq k 0)\n    (repeat n\n      (setq x (+ a (* (+ k 0.5) h)))\n      (setq s (+ s (MATH::INT:func x)))\n      (setq k (1+ k))\n    )\n    (setq t2 (\/ (+ t1 (* h s)) 2.))\n    (setq p (abs (- t1 t2)))\n    (setq t1 t2)\n    (setq n (+ n n))\n    (setq h (\/ h 2))\n    (setq iter (1+ iter))\n  )\n  t2\n)<\/pre>\n
[\/codesyntax]

\u4e5d\u3001\u6b65\u957f\u79ef\u5206\u6cd5\u3002
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u6b65\u957f\u79ef\u5206\u6cd5                                                  \n;;; Trapezoidal Integration 2                                   \n;;;=============================================================\n(defun MATH::INT:Trapzd (a b n \/ DEL IT SUM TNM X s)\n  (if (= n 1)\n    (setq s (* 0.5 (- b a) (+ (MATH::INT:func a) (MATH::INT:func b))))\n    (progn\n      (setq it 1)\n      (repeat (- n 2)\n\t(setq it (lsh it 1))\n      )\n      (setq tnm it)\n      (setq del (\/ (- b a) tnm))\n      (setq x (+ a (* 0.5 del)))\n      (setq sum 0.0)\n      (repeat it\n\t(setq sum (+ sum (MATH::INT:func x)))\n\t(setq x (+ x del))\n      )\n      (setq s (* 0.5 (+ s (\/ (* (- b a) sum) tnm))))\n    )\n  )\n)<\/pre>\n
[\/codesyntax]

\u5341\u3001\u81ea\u9002\u5e94\u79ef\u5206\u6cd5
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u81ea\u9002\u5e94\u6c42\u79ef\u5206\u6cd5                                              \n;;; Self-adapting Trapezia Integration                          \n;;;=============================================================\n(defun MATH::INT:Atrapezia (a b eps \/ F0 F1 H S T0 TT d)\n  (setq d 1e-4)\n  (setq h (- b a))\n  (setq TT '(0. . 0.))\n  (setq f0 (MATH::INT:func a))\n  (setq f1 (MATH::INT:func b))\n  (setq t0 (* h (+ f0 f1) 0.5))\n  (car (MATH::INT:ppp a b h f0 f1 t0 eps d tt))\n)\n\n(defun MATH::INT:PPP (x0 x1 h f0 f1 t0 eps d tt \/ EPS1 F G P T1 T2 T3 X X2)\n  (setq x (+ x0 (* h 0.5)))\n  (setq f (MATH::INT:func x))\n  (setq t1 (* h (+ f0 f) 0.25))\n  (setq t2 (* h (+ f1 f) 0.25))\n  (setq p (abs (- t0 t1 t2)))\n  (if (or (< p eps) (< (* 0.5 h) d))\n    (cons (+ (car tt) t1 t2) (cdr tt))\n    (progn\n      (setq g (* h 0.5))\n      (setq eps1 (\/ eps 1.4))\n      (setq t3 (MATH::INT:ppp x0 x g f0 f t1 eps1 d tt))\n      (setq t3 (MATH::INT:ppp x x1 g f f1 t2 eps1 d t3))\n    )\n  )\n)<\/pre>\n
[\/codesyntax]
\u4e00\u4e9b\u76f8\u5173\u51fd\u6570\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u529f\u80fd: \u7b2c\u4e00\u7c7b\u692d\u5706\u79ef\u5206                                        \n;;; \u8f93\u5165: Phi \u548c k < 1                                          \n;;; \u8f93\u51fa: \u6240\u6c42\u692d\u5706\u79ef\u5206\u503c                                        \n;;;=============================================================\n(defun Math:Elliptic_Integral_1 (phi kCoff \/ )\n  (defun MATH::INT:func (x \/ s)\n    (setq s (* kCoff (sin x)))\n    (\/ 1.0 (sqrt (* (- 1 s) (1+ s))))\n  )\n  (MATH::INT:Romberg 0 phi 1e-15)\n)\n\n;;;=============================================================\n;;; \u529f\u80fd: \u7b2c\u4e8c\u7c7b\u692d\u5706\u79ef\u5206                                        \n;;; \u8f93\u5165: Phi \u548c k < 1                                          \n;;; \u8f93\u51fa: \u6240\u6c42\u692d\u5706\u79ef\u5206\u503c                                        \n;;;=============================================================\n(defun Math:Elliptic_Integral_2 (phi kCoff \/ )\n  (defun MATH::INT:func (x \/)\n    (setq x (* kCoff (sin x)))\n    (sqrt (* (- 1 x) (1+ x)))\n  )\n  (MATH::INT:Romberg 0 phi 1e-15)\n)\n\n;;;=============================================================\n;;; \u9636\u4e58                                                        \n;;;=============================================================\n(defun Math::frac (n)\n  (if (= n 0)\n    1.\n    (if\t(= n 1)\n      1.\n      (* n (Math::frac (1- n)))\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u4f3d\u739b\u51fd\u6570                                                    \n;;;=============================================================\n(defun Math::gammln (xx \/ x y tmp ser j cof)\n  (setq\tcof '(76.18009172947146\t\t -86.50532032941677\n\t      24.01409824083091\t\t -1.231739572450155\n\t      0.1208650973866179e-2\t -0.5395239384953e-5\n\t     )\n  )\n  (setq x xx)\n  (setq y x)\n  (setq tmp (+ x 5.5))\n  (setq tmp (- tmp (* (+ 0.5 x) (log tmp))))\n  (setq ser 1.000000000190015)\n  (setq j 0)\n  (repeat 6\n    (setq y (1+ y))\n    (setq ser (+ ser (\/ (nth j cof) y)))\n    (setq j (1+ j))\n  )\n  (- (log (\/ (* 2.5066282746310005 ser) x)) tmp)\n)\n\n;;;=============================================================\n;;; \u6b21\u51fd\u6570\u7528\u4e8e\u6bd4\u8f83\u6392\u5e8f                                          \n;;;=============================================================\n(defun MATH::INT:funcSort (e1 e2)\n  (< (car e1) (car e2))\n)\n\n;;;=============================================================\n;;; \u7ec4\u5408\u4e24\u4e2a\u7cfb\u6570                                                \n;;;=============================================================\n(defun MATH::INT:Bind (xi wi)\n  (setq xi (vl-sort xi 'MATH::INT:funcSort))\n  (setq wi (vl-sort wi 'MATH::INT:funcSort))\n  (setq xi (mapcar 'cdr xi))\n  (setq wi (mapcar 'cdr wi))\n  (mapcar 'cons xi wi)\n)<\/pre>\n
[\/codesyntax]


\u4e0b\u9762\u662f\u7528\u5bf9\u8bdd\u6846\u521b\u5efa\u7684\u6c42\u79ef\u5206\u6cd5\u7684\u4f8b\u7a0b\uff1a

[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u7528\u5404\u79cd\u65b9\u6cd5\u6c42\u79ef\u5206\u7684\u7a0b\u5e8f                                      \n;;;=============================================================\n(defun C:Quadrature (\/ ID OK DCL_FILE)\n  (setq id (load_dialog (setq Dcl_File (MATH::INT:Write_Dcl))))\t;\u4ece\u5bf9\u8bdd\u6846\u4e2d\u5f97\u5230\u8868\u8fbe\u5f0f\n  (vl-file-delete Dcl_File)\t\t\t\t\t;\u5220\u9664\u4e34\u65f6\u5bf9\u8bdd\u6846\u6587\u4ef6\n  (setq ok 2)\n  (if (new_dialog \"dcl_Integration\" id)\n    (progn\n      (VL-CATCH-ALL-APPLY 'MATH::INT:GetSettings)\t\t;\u8bfb\u53d6\u9ed8\u8ba4\u6570\u636e\n      (action_tile \"help\" \"(MATH::INT:Help 1)\")\t\t\t;\u5e2e\u52a9\n      (foreach k '(0 1 2 3 4 5 6 7 8)\n\t(setq k (strcat \"K\" (itoa k)))\n        (action_tile k \"(MATH::INT:OnBtn $key)\")                ;\u6309\u94ae\u52a8\u4f5c\uff0c\u5bf9\u5e94\u76f8\u5bf9\u7684\u79ef\u5206\u65b9\u6cd5\n      )\t\t   \n      (setq ok (start_dialog))\n    )\n  )\n  (unload_dialog ID)\n  (princ)\n)\n\n(defun C:JF (\/)\n  (VL-CATCH-ALL-APPLY 'C:Quadrature)\n  (princ)\n)\n\n;;;=============================================================\n;;; \u4ece\u73af\u5883\u53d8\u91cf\u8bfb\u53d6\u4e0a\u6b21\u6570\u636e                                      \n;;;=============================================================\n(defun MATH::INT:GetSettings (\/ data)\n  (if (setq Data (getenv \"Intergration\"))\n    (foreach k (read data)\n      (set_tile (car k) (cdr k))\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u68c0\u67e5\u5bf9\u8bdd\u6846\u8f93\u5165                                              \n;;;=============================================================\n(defun MATH::INT:CheckInput (symS symA symB symN \/ e f)\n  (setq e (exp 1))\n  (set symS (get_tile \"F\"))\n  (set symA (MATH::INT:MyRead (get_tile \"A\")))\n  (set symB (MATH::INT:MyRead (get_tile \"B\")))\n  (set symN (MATH::INT:MyRead (get_tile \"N\")))\n  (setq f (CAL:Expr2Func (eval s) 'MATH::INT:func '(x)))\n  (apply 'and (mapcar 'eval '(F symS symA symB symN)))   \n)\n\n\n;;;=============================================================\n;;; \u6309\u94ae\u52a8\u4f5c\uff0c\u5bf9\u5e94\u76f8\u5e94\u7684\u51fd\u6570\u6c42\u79ef\u5206                              \n;;;=============================================================\n(defun MATH::INT:OnBtn (key \/ DATA s N a b OldZIN m EPS RET tm0 map msg foo tmp idx)\n  (setq m (VL-CATCH-ALL-APPLY 'MATH::INT:CheckInput '(s a b n)))\n  (if (or (vl-catch-all-error-p m) (not m) (equal a b 1e-8))\n    (if (vl-catch-all-error-p m) \n      (set_tile \"info\" (vl-catch-all-error-message m))\n      (set_tile \"info\" \"\u65e0\u6548\u8f93\u5165!\")\n    )\n    (progn\n      ;;\u5982\u679c\u7cbe\u5ea6\u8fc7\u9ad8\uff0c\u8bbe\u7f6e\u4e3a15\u4f4d\u7684\u7cbe\u5ea6\n      (if (> n 20)\n\t(setq n 15)\n\t(setq n (fix (abs n)))\n      )\n      ;;\u5982\u679c\u4e0a\u533a\u95f4\u5c0f\u4e8e\u4e0b\u533a\u95f4\uff0c\u5219\u4ea4\u6362\u533a\u95f4\n      (if (< b a)\n\t(setq tmp a a b b tmp)\n      )\n      ;;\u8bb0\u4f4f\u5bf9\u8bdd\u6846\u8f93\u5165\uff0c\u7528\u4e8e\u4e0b\u6b21\n      (setq OldZIN (getvar \"DIMZIN\"))\n      (setvar \"DIMZIN\" 8)\n      (set_tile \"N\" (itoa n))\n      (set_tile \"A\" (rtos a 2 20))\n      (set_tile \"B\" (rtos b 2 20))\n      (setq data (list (cons \"F\" s)\n\t\t       (cons \"N\" (itoa n))\n\t\t       (cons \"A\" (rtos a 2 20))\n\t\t       (cons \"B\" (rtos b 2 20))\n\t\t )\n      )\n      (setvar \"DIMZIN\" OldZIN)\n      (setenv \"Intergration\" (VL-PRIN1-TO-STRING data))\n      ;;\u5f00\u59cb\u8ba1\u7b97\u79ef\u5206\n      (setq eps (expt 0.1 n))\n      (setq tm0 (getvar \"TDUSRTIMER\"))\n      (setq map (MATH::INT:GetMethods))\t\t\t\t;\u79ef\u5206\u8ba1\u7b97\u65b9\u6cd5\u96c6\n      (setq idx (atoi (substr key 2)))\t\t\t\t\n      (setq foo (nth idx map))\t\t\t\t\t;\u83b7\u53d6\u8ba1\u7b97\u79ef\u5206\u7684\u51fd\u6570\n      (setq ret (VL-CATCH-ALL-APPLY foo (list a b eps)))        ;\u83b7\u53d6\u79ef\u5206\u503c\n      (if (vl-catch-all-error-p ret)\n\t(set_tile \"info\" (vl-catch-all-error-message ret))\t;\u6c42\u89e3\u8fc7\u7a0b\u53d1\u751f\u4e86\u9519\u8bef\n        (if (null ret)\n\t  (set_tile \"info\" \"\u53d1\u751f\u4e86\u9519\u8bef\uff0c\u6c42\u503c\u7ed3\u679c\u4e3a\u7a7a!\")\t\n\t  (progn\n\t    ;(MATH::INT:Bench 100 a b eps)\n            (setq ret (rtos ret 2 20))\n            (setq msg (get_attr key \"label\"))\n            (setq msg (strcat msg \"\u6c42\u7684\u7ed3\u679c\u4e3a:\" ret))\n            (set_tile (strcat \"R\" (itoa idx)) ret)             \t;\u663e\u793a\u6c42\u89e3\u7ed3\u679c\n            (set_tile \"info\" msg)\t\t\t\t;\u663e\u793a\u6c42\u89e3\u7ed3\u679c\n            (princ (strcat \"\\n\" msg))\t\t\t\t;\u6253\u5370\u6c42\u89e3\u7ed3\u679c\n            (princ \"\\n\u8d39\u65f6:\")\n            (princ (* (- (getvar \"TDUSRTIMER\") tm0) 86400))\n            (princ \"\u79d2.\")\n\t  )\n        )\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u5404\u79cd\u79ef\u5206\u6d4b\u901f                                                \n;;;=============================================================\n(defun MATH::INT:Bench (n a b eps)\n  (UTI:BENCH\n    n\n    (list\n      (list 'MATH::INT:Romberg a b eps)\n      (list 'MATH::INT:Gauss-Legendre a b eps)\n      (list 'MATH::INT:Simpson a b eps)\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u79ef\u5206\u65b9\u6cd5\u96c6                                                  \n;;;=============================================================\n(defun MATH::INT:GetMethods ()\n  '(MATH::INT:Romberg\n    MATH::INT:Simpson\n    MATH::INT:Atrapezia\n    MATH::INT:Trapezia\n    MATH::INT:Gauss-Legendre\n    Math::INT:Gauss-Chebyshev\n    Math::INT:Gauss-Laguerre\n    Math::INT:Gauss-Hermite\n    MATH::INT:Gauss-Jacobi\n   )\n)\n\n;;;=============================================================\n;;; \u8868\u8fbe\u5f0f\u6c42\u503c\uff0c\u4e5f\u53ef\u4ee5\u7528cal\u51fd\u6570                                 \n;;;=============================================================\n(defun MATH::INT:MyRead (str \/ e)\n  (setq e (exp 1))\n  (CAL:Expr2Value str)\n)\n\n;;;=============================================================\n;;; \u5e2e\u52a9\u548c\u8bf4\u660e: help and instruction                            \n;;;=============================================================\n(defun MATH::INT:Help (n)\n  (if (= n 1)\n    (if\t(= \"CHS\" (getvar \"Locale\"))\n      (alert\n\t\"\u51fd\u6570\u5f0f\u53ea\u63a5\u53d7\u7b26\u53f7x\u4e3a\u53d8\u91cf,\u4e0d\u89c4\u8303\u5f88\u53ef\u80fd\u51fa\u9519!\n\t\\n\u51fd\u6570\u53ef\u4ee5LISP\u5185\u7f6e\u7684\u6570\u5b66\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u81ea\u5b9a\u4e49\u51fd\u6570!\n\t\\n\u6307\u6570\u7528^\u8868\u793a\uff0c+-*\/\u8868\u793a\u52a0\u51cf\u4e58\u9664\uff0c\u4e58\u53f7\u4e0d\u80fd\u7701\u7565\u3002\n\t\\n\u7a0b\u5e8f\u80fd\u91c7\u7528\u591a\u79cd\u65b9\u6cd5\u6c42\u79ef,\u4e00\u822c\u6765\u8bf4\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\u6700\u5feb\u3002\n\t\\n\u6709\u4ec0\u4e48\u95ee\u9898email: highflybird@qq.com\n\t\\n\u4f5c\u8005: highflybird \u65e5\u671f2019.07\"\n      )\n      (alert\n\t\"Standard expression only accepts \\\"x\\\" as a variale!\n\t\\nThe fastest is Romberg Integration,the slowest is Trapezoidal rule(Be careful!).\n\t\\nRecommendation:Don't set a high precision at first,promote it step by step.\n\t\\nEspically for the Trapezoidal rule, It won't work well on some circumstances.\n\t\\nIt's an Open Source Software. Thanks for your advice or bug reports.\n\t\\nAuthor: highflybird  Email: highflybird@qq.com  Date:2019.07.\"\n      )\n    )\n    (set_tile \"info\" \"\u8868\u8fbe\u5f0f\u975e\u6cd5\u6216\u8005\u7a7a\u8f93\u5165.\")\n  )\n)\n<\/pre>\n
[\/codesyntax]
\u5bf9\u8bdd\u6846\u7684\u5236\u4f5c\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u8f93\u5165\u5bf9\u8bdd\u6846                                                  \n;;;=============================================================\n(defun UTI:Inputbox (\/ str wcs ret)\n  (setq\tstr \"Function GetNumbers()\n  \t     GetNumbers=inputbox(\\\"\u8bf7\u8f93\u5165\u4e24\u4e2a\u53c2\u6570,\u4e2d\u95f4\u7528\u7a7a\u683c\u9694\u5f00:\\\",\\\"\u8f93\u5165\u6846\\\")\n             End Function\"\n  )\n  (if\n    (or\n      (setq wcs (vlax-create-object \"Aec32BitAppServer.AecScriptControl.1\"))\n      (setq wcs (vlax-create-object \"ScriptControl\"))\n    )\n    (progn \n      (vlax-put-property wcs \"language\" \"VBScript\")\n      (vlax-invoke wcs 'addcode str)\n      (if (setq ret (vlax-invoke wcs 'run \"GetNumbers\"))\n\t(setq ret (strcat \"(\" ret \")\")\n\t      ret (read ret)\n\t)\n      )\n      (vlax-release-object wcs)\n      (if\n\t(and\n\t  (= 2 (length ret))\n\t  (or (= 'INT (type (car ret))) (= 'REAL (type (car ret))))\n\t  (or (= 'INT (type (cadr ret))) (= 'REAL (type (cadr ret))))\n\t)\n\tret\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u5199\u5bf9\u8bdd\u6846\u5230\u6587\u4ef6\u7528\u4e8e\u7a0b\u5e8f                                      \n;;;=============================================================\n(defun MATH::INT:Write_Dcl (\/ Dcl_File file str)\n  (setq Dcl_File (vl-filename-mktemp nil nil \".Dcl\"))\n  (setq file (open Dcl_File \"w\"))\n  (princ\n    \"dcl_Integration : dialog {\n\tlabel = \\\"\u6570\u503c\u79ef\u5206LISP\u7248  v1.2\\\";\n\t: boxed_column {\n          width = 60;\n\t  fixed_width = true;\n\t  : edit_box {\n\t    key=\\\"F\\\";\n\t    label= \\\"\u51fd\u6570:\\\";\n\t  }\n\t  : row {\n\t    : edit_box {\n\t      key=\\\"A\\\";\n\t      label= \\\"\u4e0b\u9650:\\\";\n\t    }\n\t    : edit_box {\n\t      key=\\\"B\\\";\n\t      label= \\\"\u4e0a\u9650:\\\";\n\t    }      \n\t    : edit_box {\n\t      key=\\\"N\\\";\n\t      label = \\\"\u7cbe\u786e\u4f4d\u6570:\\\";\n\t      value = 8;\n\t      edit_width = 2;\n\t      fixed_width = true;\n\t    }\n\t  }\n\t  spacer_1;\n\t}\n\t: row {\n\t  : boxed_column {\n\t    label = \\\"\u8ba1\u7b97\u65b9\u6cd5:\\\";\n\t    : button {\n\t      key = \\\"K0\\\";\n\t      label = \\\"\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K1\\\";\n\t      label = \\\"\u8f9b\u666e\u68ee\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K2\\\";\n\t      label = \\\"\u81ea\u9002\u5e94\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K3\\\";\n\t      label = \\\"\u53d8\u6b65\u957f\u68af\u5f62\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K4\\\";\n\t      label = \\\"\u9ad8\u65af-\u52d2\u8ba9\u5fb7\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K5\\\";\n\t      label = \\\"\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K6\\\";\n\t      label = \\\"\u9ad8\u65af-\u62c9\u76d6\u5c14\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K7\\\";\n\t      label = \\\"\u9ad8\u65af-\u57c3\u5c14\u7c73\u7279\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K8\\\";\n\t      label = \\\"\u9ad8\u65af-\u96c5\u514b\u6bd4\u6cd5\\\";\n\t    }\n\t    spacer;\n\t  }\n\t  : boxed_column {\n\t    width = 32;\n\t    fixed_width = true;\n\t    label = \\\"\u8ba1\u7b97\u7ed3\u679c:\\\";\n\t    : text {\n\t      key = \\\"R0\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R1\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R2\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R3\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R4\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R5\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R6\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R7\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R8\\\";\n\t    }\n\t    spacer;\n\t  }\n\t}\n\tok_cancel_help;\n\t\/\/ok_cancel_help_errtile;\n\t: text {\n\t  key = \\\"info\\\";\n\t  label = \\\"Copyright \\\\u+00A9 2007-2019 Highflybird. All rights reserved.\\\";\n\t  width = 20;\n\t}\n    }  \"\n    file\n  )\n  (close file)\n  Dcl_File\n)\n\n;;;=============================================================\n;;; \u4e0b\u9762\u7684\u5c31\u4e0d\u7528\u4ecb\u7ecd\u4e86                                          \n;;;=============================================================\n(vl-load-com)\n(if (= \"CHS\" (getvar \"Locale\"))\n  (prompt \"\u8f93\u5165\u547d\u4ee4: JF\")\n  (prompt \"Please enter: Quadrature\")\n)\n(c:JF)\n(princ)<\/pre>\n
[\/codesyntax]

<\/pre>\n\n\n
\u4e00\u4e9b\u5e94\u7528\uff0c\u5982\u4e0b\u9762\u7f51\u53cb\u7684\u63d0\u95ee\u5e76\u89e3\u7b54\uff1a<\/p>\n\n\n\n
http:\/\/bbs.mjtd.com\/forum.php?mod=viewthread&tid=179809&extra=&highlight=%BB%FD%B7%D6&page=1<\/a><\/p>\n\n\n\n
\n\n\u692d\u7403\u4f53\u7403\u7f3a\u7684\u8868\u9762\u79ef\u8ba1\u7b97
\u7528\u79ef\u5206<\/strong>\u516c\u5f0f\u7528LISP\u7a0b\u5e8f\u8868\u8fbe\u5e76\u8ba1\u7b97\u51fa\u6765
\u8fd9\u662f\u4e00\u4e2a\u7f51\u4e0a\u7684
\u6709\u7ed3\u679c\u548c\u516c\u5f0f\n\n<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\u7528\u4e0a\u9762\u7684\u4ecb\u7ecd\u7684\u4e00\u4e9b\u51fd\u6570\u53ef\u4ee5\u5f97\u51fa\u5176\u9762\u79ef\u548c\u4f53\u79ef\uff1a<\/p>\n\n\n\n
\u7528LISP\u7f16\u5199\u4e86\u4e00\u4e2a\u6c42\u79ef\u5206\u7684\u7a0b\u5e8f\uff1a<\/p>\n\n\n\n
 \u91cc\u9762\u91c7\u7528\u4e86\u5404\u79cd\u65b9\u6cd5\u6c42\u79ef\u5206\u548c\u5404\u79cd\u7c7b\u578b\u7684\u79ef\u5206\u3002\u4e0b\u9762\u6211\u628a\u5404\u79cd\u65b9\u6cd5\u7684\u6e90\u7801\u8d34\u51fa\u3002 <\/p>\n\n\n\n
\u65b9\u6cd5\u4e00\uff1a \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u79ef\u5206\u6cd5\u3002<\/p>\n\n\n
[codesyntax lang=”lisp”]<\/p>\n
;;;=============================================================\n;;; \u8ba1\u7b97\u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570\u7684\u7cfb\u6570 \uff0c\u5982\u4e0b\u9762\u76846\u6b21\u9879\u7cfb\u6570           \n;;; p0(x) = 1                                                   \n;;; p1(x) = x                                                   \n;;; p2(x) = (3*x^2-1)\/2                                         \n;;; p3(x) = (5*x^3-3*x)\/2                                       \n;;; p4(x) = (35*x^4-30*x^2+3)\/8                                 \n;;; p5(x) = (63*x^5-70*x^3+15*x)\/8                              \n;;; p6(x) = (231*x^6-315*x^4+105*x^2-5)\/16                      \n;;; x-2                                                         \n;;; x-1                                                         \n;;; 9x-5                                                        \n;;; 216*x^2-216*x+49                                            \n;;; 45000*x^2-32200*x+5103                                      \n;;; 2025000*x^3-2025000*x^2+629325*x-58564                      \n;;; 142943535000*x^3-1130712534000*x^2+27510743799*x-1976763932 \n;;; \u8f93\u5165: x1,x2 \u533a\u95f4(\u4e00\u822c\u6765\u8bf4\u662f-1..1)\uff0cn \u8fed\u4ee3\u6b21\u6570,eps\u8fed\u4ee3\u7cbe\u5ea6   \n;;; \u8f93\u51fa: \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570\u7684\u7cfb\u6570,\u7528\u70b9\u8868\u96c6\u8868\u793a                \n;;; http:\/\/mathworld.wolfram.com\/Legendre-GaussQuadrature.html  \n;;;=============================================================\n(defun Math::Int:Legendre_Polynomial (x1 x2 n eps \/ xi wi FI I ITER J M P1 P2 P3 PP XL XM Z Z1)\n  (setq m (\/ (1+ n) 2))\n  (setq xm (* 0.5 (+ x2 x1)))\n  (setq xl (* 0.5 (- x2 x1)))\n  (setq i 1)\n  (repeat m\n    (setq z (cos (\/ (* pi (- i 0.25)) (+ n 0.5))))\n    (setq iter 0)\n    (while (and (not (equal z z1 eps)) (< iter 1000))\n      (setq p1 1.0)\n      (setq p2 0.0)\n      (setq j 1)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (\/ (- (* z p2 (+ j j -1.)) (* (1- j) p3)) j))\n\t(setq j (1+ j))\n      )\n      (setq pp (\/ (* n (- (* z p1) p2)) (1- (* z z))))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq iter (1+ iter))\n    )\n    (setq fi (\/ xl 0.5 (- 1 (* z z)) pp pp))\n    (setq xi (cons (cons (1- i) (- xm (* xl z))) xi))\n    (setq wi (cons (cons (1- i) fi) wi))\n    (if (\/= (1- i) (- n i))\n      (setq xi (cons (cons (- n i) (+ xm (* xl z))) xi)\n\t    wi (cons (cons (- n i) fi) wi)\n      )\n    )\n    (setq i (1+ i)) \n  )\n  (MATH::INT:Bind xi wi)\n)\n\n;;;=============================================================\n;;; \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570                                         \n;;;=============================================================\n(defun MATH::INT:Gauss-Legendre (a b eps \/ AA BB EP FX G H I L M P S W X)\n  (setq l '((-0.93246951420315202787 . 0.17132449237917034504)  \n\t    (-0.66120938646626451363 . 0.36076157304813860756)     \n\t    (-0.23861918608319690859 . 0.46791393457269104739)  \n\t    ( 0.23861918608319690859 . 0.46791393457269104739)     \n            ( 0.66120938646626451363 . 0.36076157304813860756)     \n\t    ( 0.93246951420315202787 . 0.17132449237917034504)\n\t  )\n  )                                                         \n  ;(setq L (Math::Int:Legendre_Polynomial -1 1 100 2e-20))\n  (setq m 1)\n  (setq h (- b a))\n  (setq s (abs (* 0.001 h)))\n  (setq p 1e100) \t\n  (setq ep (1+ eps))\n  (while (and (>= ep eps) (> (abs h) s))\n    (setq g 0)\n    (setq i 1)\n    (repeat m\n      (setq bb (+ a (* i h)))\n      (setq aa (- bb h))\n      (setq w 0)\n      (foreach k l\n\t(setq x (* 0.5 (+ bb aa (* (- bb aa) (car k)))))\n\t(setq fx (MATH::INT:func x))\n\t(setq w (+ w (* fx (cdr k))))\n      )\n      (setq g (+ g w))\n      (setq i (1+ i))\n    ) \n    (setq g (* g h 0.5))\n    (setq ep (\/ (abs (- g p)) (1+ (abs g))))\n    (setq p g)\n    (setq m (1+ m))\n    (setq h (\/ (- b a) m))\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u52d2\u8ba9\u5fb7-\u9ad8\u65af\u6c42\u79ef\u51fd\u6570(\u53e6\u4e00\u65b9\u6cd5\uff0c\u6162\u4e9b)                         \n;;;=============================================================\n(defun MATH::INT:Gauss-Legendre1 (a b eps \/ FLAG G H L N X Y)\n  (setq n 1)                                                      \n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (while (and (< n 100) flag)\t\n    (setq g 0)\n    (setq L (Math::Int:Legendre_Polynomial a b n eps))\n    (foreach w L\n      (setq x (car w))\n      (setq y (MATH::INT:func x))\n      (setq g (+ g (* (cdr w) y)))\n    )\n    (if (equal g h eps)\n      (setq flag nil)\n      (setq n (+ n n)\n\t    h g\n      )\n    )\n  )\n  g\n)[\/codesyntax]<\/pre>\n\n\n
\u65b9\u6cd5\u4e8c\uff1a\u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206<\/p>\n\n\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206                                           \n;;; \u529f\u80fd: \u8ba1\u7b97 e^(-x^2)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f4-INF..INF\u4e0a)       \n;;;=============================================================\n(defun Math::INT:Gauss-Hermite (a b eps \/ n L g x y)\n  (setq n 100)\n  (setq L (Math::INT:GetHermite n))\n  (setq g 0)\n  (foreach w L\n    (setq x (car w))\n    (setq y (MATH::INT:func x))\n    (setq g (+ g (* (cdr w) y)))\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u83b7\u53d6\u57c3\u7c73\u5c14\u7279\u7cfb\u6570                                            \n;;;=============================================================\n(defun Math::INT:GetHermite (n \/ xi wi EPS FI I ITS J M MAXIT P1 P2 P3 PIM4 PP Z Z1)\n  (setq eps 1e-15)\n  (setq PIM4 0.7511255444649425)\n  (setq maxIt 10)\n  (setq m (\/ (1+ n) 2))\n  (setq i 0)\n  (while (< i m)\n    (if (= i 0)\n      (setq z (- (sqrt (+ n n 1)) (* 1.85575 (expt (+ n n 1) (\/ -1 6.)))))\n      (if (= i 1)\n\t(setq z (- z (\/ (* 1.14 (expt n 0.426)) z)))\n\t(if (= i 2)\n\t  (setq z (- (* 1.86 z) (* 0.86 (cdr (assoc 0 xi)))))\n\t  (if (= i 3)\n\t    (setq z (- (* 1.91 z) (* 0.91 (cdr (assoc 1 xi)))))\n\t    (setq z (- (+ z z) (cdr (assoc (- i 2) xi))))\n\t  )\n\t)\n      )\n    )\n    (setq its 0)\n    (while (and (not (equal z z1 eps)) (< its MAXIT))\n      (setq p1 pIM4)\n      (setq p2 0.0)\n      (setq j 0)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (- (* z p2 (sqrt (\/ 2.0 (1+ j)))) (* p3 (sqrt (\/ j (+ 1.0 j))))))\n\t(setq j (1+ j))\n      )\n      (setq pp (* p2 (sqrt (+ n n))))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq its (1+ its))\n    )\n    (setq fi (\/ 2.0  pp pp))\n    (setq xi (cons (cons i z) xi))\n    (setq wi (cons (cons i fi) wi))\n    (if (\/= i (- n 1 i))\n      (setq xi (cons (cons (- n 1 i) (- z)) xi)\n\t    wi (cons (cons (- n 1 i) fi) wi)\n      )\n    )\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)[\/codesyntax]
<\/pre>\n\n\n
\u65b9\u6cd5\u4e09\uff1a\u9ad8\u65af-\u57c3\u7c73\u5c14\u7279\u79ef\u5206<\/p>\n\n\n\n
\u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206<\/p>\n\n\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97 f(x)*((1-x)^a)*((1+x)^b)\u7684\u79ef\u5206(\u5728\u533a\u95f4-1..1\u4e0a)    \n;;;=============================================================\n(defun MATH::INT:Gauss-Jacobi (a b eps \/ ALF ARGS BET G N X Y flag g0)\n  (if (setq args (UTI:InputBox))\n    (progn\n      (setq flag T)\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n      (setq n 10)\n      (while (and (< n 100) flag)\t\n        (setq alf (car args))\n        (setq bet (cadr args))\n        (setq g 0)\n        (foreach w (MATH::INT:GetJacobiPolynomial n alf bet)\n          (setq x (car w))\n          (setq y (MATH::INT:func x))\n          (setq g (+ g (* (cdr w) y)))\n        )\n\t(if (equal g g0 eps)\n\t  (setq flag nil g0 g)\n\t  (setq n (+ n n) g0 g)\n\t)\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97\u9ad8\u65af-\u96c5\u514b\u6bd4\u79ef\u5206\u7cfb\u6570\u9879                             \n;;;=============================================================\n(defun MATH::INT:GetJacobiPolynomial (n alf bet \/\n\t\t\t\t      xi wi A ALFBET AN B BN C EPS FI FLG I ITS\n\t\t\t\t      J MAXIT P1 P2 P3 PP R1 R2 R3 TEMP Z Z1)\n  (setq maxIT 40)\n  (setq eps 1e-15)\n  (setq alf (float alf))\n  (setq bet (float bet))\n  (setq i 0)\n  (repeat n\t\t\t\t\t\t\t\t\t;for (i=0;i<n;i++) {\n    (cond\n      ( (= i 0)\t\t\t\t\t\t\t\t\t;if (i == 0) {\n        (setq an (\/ alf n))\t\t\t\t\t\t\t;an=alf\/n;\n        (setq bn (\/ bet n))\t\t\t\t\t\t\t;bn=bet\/n;\n        (setq r1 (* (1+ alf) (+ (\/ 2.78 (+ 4.0 (* n n))) (\/ (* 0.768 an) n))))\t;r1=(1.0+alf)*(2.78\/(4.0+n*n)+0.768*an\/n);\n        (setq r2 (+ 1.0 (* 1.48 n) (* 0.96 bn) (* 0.452 an an) (* 0.83 an bn)))\t;r2=1.0+1.48*an+0.96*bn+0.452*an*an+0.83*an*bn;\n        (setq z  (- 1 (\/ r1 r2)))\t\t\t\t\t\t;z=1.0-r1\/r2;\n      )\n      ( (= i 1)\t\t\t\t\t\t\t\t\t;else if (i == 1)\n        (setq r1 (\/ (+ 4.1 alf) (* (1+ alf) (1+ (* 0.156 alf)))))\t\t;r1=(4.1+alf)\/((1.0+alf)*(1.0+0.156*alf));\n        (setq r2 (1+ (\/ (* 0.06 (- n 8.0) (1+ (* 0.12 alf))) n)))\t\t;r2=1.0+0.06*(n-8.0)*(1.0+0.12*alf)\/n;\n        (setq r3 (1+ (\/ (* 0.012 bet (1+ (* 0.25 (abs alf)))) n)))\t\t;r3=1.0+0.012*bet*(1.0+0.25*fabs(alf))\/n;\n        (setq z  (- z (* (- 1 z) r1 r2 r3)))\t\t\t\t\t;z -= (1.0-z)*r1*r2*r3;\n      )\n      ( (= i 2)\t\t\t\t\t\t\t\t\t;else if (i == 2)\n        (setq r1 (\/ (+ 1.67 (* 0.28 alf)) (1+ (* 0.37 alf))))\t\t\t;r1=(1.67+0.28*alf)\/(1.0+0.37*alf);\n        (setq r2 (1+ (\/ (* 0.22 (- n 8.0)) n)))\t\t\t\t\t;r2=1.0+0.22*(n-8.0)\/n;\n        (setq r3 (1+ (\/ (* 8.0 bet) (* (+ 6.28 bet) n n))))\t\t\t;r3=1.0+8.0*bet\/((6.28+bet)*n*n);\n        (setq z  (- z (* (- (cdr (assoc 0 xi)) z) r1 r2 r3)))\t\t\t;z -= (x[0]-z)*r1*r2*r3;\n      )\n      ( (= i (- n 2))\t\t\t\t\t\t\t\t;else if (i == n-2)\n        (setq r1 (\/ (1+ (* 0.235 bet)) (+ 0.766 (* 0.119 bet))))\t\t;r1=(1.0+0.37*bet)\/(1.67+0.28*bet);\n        (setq r2 (\/ 1.0 (1+ (\/ (* 0.639 (- n 4.0)) (1+ (* 0.71 (- n 4.0)))))))\t;r2=1.0\/(1.0+0.639*(n-4.0)\/(1.0+0.71*(n-4.0)));\n        (setq r3 (\/ 1.0 (1+ (\/ (* 20.0 alf) (* (+ 7.5 alf) n n))))) \t\t;r3=1.0\/(1.0+20.0*alf\/((7.5+alf)*n*n));\n        (setq z  (+ z (* (- z (cdr (assoc (- n 4) xi))) r1 r2 r3))) \t\t;z += (z-x[n-4])*r1*r2*r3;\n      )\n      ( (= i (1- n))\t\t\t\t\t\t\t\t;else if (i == n-1)\n        (setq r1 (\/ (1+ (* 0.37 bet))(+ 1.67 (* 0.28 bet))))\t\t\t;r1=(1.0+0.37*bet)\/(1.67+0.28*bet);\n        (setq r2 (\/ 1.0 (1+ (\/ (* 0.22 (- n 8.0)) n))))\t\t\t\t;r2=1.0\/(1.0+0.22*(n-8.0)\/n);\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));\n       \t(setq z  (+ z (* (- z (cdr (assoc (- n 3) xi))) r1 r2 r3))) \t\t;z += (z-x[n-3])*r1*r2*r3;\n      )\n      (t\t\t\t\t\t\t\t\t\t;else {\n        (setq z (* 3 (- (cdr (assoc (1- i) xi)) (cdr (assoc (- i 2) xi)))))\t;z=3.0*x[i-1]-3.0*x[i-2]+x[i-3];\n        (setq z (+ z (cdr (assoc (- i 3) xi))))\n      )\n    )\n    (setq alfbet (+ alf bet))\t\t\t\t\t\t\t;alfbet=alf+bet;\n    (setq its 1)\n    (setq flg T)\n    (while (and flg (<= its maxIt))\t\t\t\t\t\t;for (its=1;its<=MAXIT;its++)\n      (setq temp (+ 2.0 alfbet))\t\t\t\t\t\t;temp=2.0+alfbet;\n      (setq p1 (\/ (+ (- alf bet) (* temp z)) 2.0))\t\t\t\t;p1=(alf-bet+temp*z)\/2.0;\n      (setq p2 1.0)\t\t\t\t\t\t\t\t;p2=1.0;\n      (setq j 2)\t\t\t\t\t\t\t\t\n      (while (<= j n)\t\t\t\t\t\t\t\t;for (j=2;j<=n;j++) {\n\t(setq p3 p2)\t\t\t\t\t\t\t\t;p3=p2;\n\t(setq p2 p1)\t\t\t\t\t\t\t\t;p2=p1;\n\t(setq temp (+ j j alfbet))\t\t\t\t\t\t;temp=2*j+alfbet;\n\t(setq a (* 2 j (+ j alfbet) (- temp 2.0)))\t\t\t\t;a=2*j*(j+alfbet)*(temp-2.0);\n\t(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);\n\t(setq c (* 2 (+ j -1 alf) (+ j -1 bet) temp))\t\t\t\t;c=2.0*(j-1+alf)*(j-1+bet)*temp;\n        (setq p1 (\/ (- (* b p2) (* c p3)) a))\t\t\t\t\t;p1=(b*p2-c*p3)\/a;\n\t(setq j (1+ j))\n      )\n      \n      (setq pp (\/ (+ (* N (- ALF BET (* TEMP Z)) P1)(* 2 (+ N ALF)(+ N BET) P2))\n\t\t  (* TEMP (- 1.0 (* Z Z)))\n\t       )\n      )\t\t\t\t\t\t\t\t\t\t;pp=(n*(alf-bet-temp*z)*p1+2.0*(n+alf)*(n+bet)*p2)\/(temp*(1.0-z*z));\n      (setq z1 z)\t\t\t\t\t\t\t\t;z1=z;\n      (setq z (- z1 (\/ p1 pp)))\t\t\t\t\t\t\t;z=z1-p1\/pp;\n      (setq its (1+ its))\n      (if (equal z z1 eps)\n\t(setq flg nil)\n      )\n    )\n    (if (> its MAXIT)\n      (princ \"\\nToo many iterations in gaujac!\")\t\t\t\t;if (its > MAXIT) nrerror(\"too many iterations in gaujac\");\n    )\n    (setq xi (cons (cons i z) xi))\t\t\t\t\t\t;x[i]=z;\n    (setq fi (exp (- (Math::gammln (+ alf n))\n\t\t     (- (Math::gammln (+ bet n)))\n\t\t     (Math::gammln (1+ n))\n\t\t     (Math::gammln (+ n alfbet 1))\n\t\t  )\n\t     )\n    )\n    (setq fi (\/ (* fi temp (expt 2.0 alfbet)) pp p2))\t\t\t\t;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);\t\n    (setq wi (cons (cons i fi) wi))\t\t\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)<\/pre>\n
[\/codesyntax]<\/p>\n
\u56db\u3001\u9ad8\u65af-\u62c9\u76d6\u5c14\u79ef\u5206 
\u529f\u80fd: \u8ba1\u7b97 e^(-x)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f40..INF\u4e0a)<\/p>\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u62c9\u76d6\u5c14\u79ef\u5206                                             \n;;; \u529f\u80fd: \u8ba1\u7b97 e^(-x)*f(x)\u7684\u5e7f\u4e49\u79ef\u5206(\u5728\u533a\u95f40..INF\u4e0a)            \n;;;=============================================================\n(defun Math::INT:Gauss-Laguerre (a b eps \/ n flag L g h x y maxN)\n  (setq n 2)\n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (setq maxN 40)\n  (while (and flag (< n maxN))\t\n    (setq g 0)\n    (setq L (Math::INT:GetLaguerre n 0))\n    (foreach w L\n      (setq x (car w))\n      (setq y (MATH::INT:func x))\n      (setq g (+ g (* (cdr w) y)))\n    )\n    (if (equal g h eps)\n      (setq flag nil)\n      (setq n (+ n n)\n\t    h g\n      )\n    )\n  )\n  g\n)\n\n;;;=============================================================\n;;; \u83b7\u53d6\u62c9\u76d6\u5c14\u7cfb\u6570                                              \n;;;=============================================================\n(defun Math::INT:GetLaguerre (n alf \/ xi wi AI EPS I ITS J MAXIT P1 P2 P3 PP X XI2 YI Z Z1)\n  (setq maxIt 10)\n  (setq eps 1e-15)\n  (setq i 0)\n  (repeat n\n    (if\t(= i 0)\n      (setq z (\/ (* (1+ alf) (+ 3 (* 0.92 alf)))\n\t\t (+ 1 (* 2.4 n) (* 1.8 alf))\n\t      )\n      )\n      (if (= i 1)\n\t(setq\n\t  z (+ z (\/ (+ 15 (* 6.25 alf)) (+ 1 (* 0.9 alf) (* 2.5 n))))\n\t)\n\t(setq ai  (1- i)\n\t      xi2 (cdr (assoc (- i 2) xi))\n\t      x\t  (\/ (*\n\t\t       (+\n\t\t\t (\/ (1+ (* 2.55 AI)) (* 1.9 AI))\n\t\t\t (\/ (* 1.26 AI ALF) (1+ (* 3.5 AI)))\n\t\t       )\n\t\t       (- Z XI2)\n\t\t     )\n\t\t     (1+ (* 0.3 ALF))\n\t\t  )\n\t      z\t  (+ z x)\n\t)\n      )\n    )\n    (setq its 0)\n    (while (and (not (equal z z1 eps)) (< its maxIt))\n      (setq p1 1)\n      (setq p2 0)\n      (setq j 0)\n      (repeat n\n\t(setq p3 p2)\n\t(setq p2 p1)\n\t(setq p1 (\/ (- (* (+ J J 1.0 ALF (- Z)) P2) (* (+ J ALF) P3))\n\t\t    (1+ J)\n\t\t )\n\t)\n\t(setq j (1+ j))\n      )\n      (setq pp (\/ (- (* n p1) (* p2 (+ n alf))) z))\n      (setq z1 z)\n      (setq z (- z1 (\/ p1 pp)))\n      (setq its (1+ its))\n    )\n    (setq xi (cons (cons i z) xi))\n    (setq yi (-\t(\/ (exp (- (Math::gammln (+ alf n)) (Math::gammln n)))\n\t\t   (* pp n p2)\n\t\t)\n\t     )\n    )\n    (setq wi (cons (cons i yi) wi))\n    (setq i (1+ i))\n  )\n  (MATH::INT:Bind xi wi)\n)
[\/codesyntax]
\u4e94\u3001\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206 
\u6b64\u65b9\u6cd5\u7528\u4e8e\u9488\u5bf9 sqrt(1-x^2)*f(x)\u578b\u7684\u79ef\u5206\u6709\u5f88\u9ad8\u7684\u6548\u7387
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206                                           \n;;; \u6b64\u65b9\u6cd5\u7528\u4e8e\u9488\u5bf9 sqrt(1-x^2)*f(x)\u578b\u7684\u79ef\u5206\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002       \n;;; \u8f93\u5165: foo \u51fd\u6570\u540d,arg \u9664\u81ea\u53d8\u91cfx\u5916\u7684\u53c2\u6570\u5217\u8868, n\u8fed\u4ee3\u6b21\u6570\u3002     \n;;; \u8f93\u51fa: \u6b64\u7c7b\u79ef\u5206\u7684\u6570\u503c.                                       \n;;; \u8bf4\u660e: \u6b64\u79ef\u5206\u6cd5\u6548\u7387\u8f83\u9ad8\uff0cn\u53d6\u503c\u4e00\u822c10\u6b21\u5de6\u53f3\u5c31\u8fbe\u5230\u6709\u6548\u6d6e\u70b9\u7cbe\u5ea6.\n;;; http:\/\/mathworld.wolfram.com\/Chebyshev-GaussQuadrature.html \n;;;=============================================================\n(defun Math::INT:Gauss-Chebyshev (a b eps \/ FI FLAG I N S0 SX WI XI)\n  (setq n 3)\t\t\t\t\t\t\t;\u4e00\u822c\u6765\u8bf4\u8fed\u4ee36-5\u6b21\u5de6\u53f3\u5c31\u53ef\u4ee5\u8fbe\u5230\u6d6e\u70b9\u8ba1\u7b97\u7cbe\u5ea6\n  (setq flag T)\t\t\t\t\t\t\t;\u662f\u5426\u8fdb\u884c\u8fed\u4ee3\n  (while (and (< n 1000) flag)\t\t\t\t\t;\u8fed\u4ee3\u6b21\u6570\u4e0d\u8d85\u8fc7100\n    (setq wi (\/ pi n))\t\t\t\t\t\t;\u6240\u6709\u7684\u6743\u503c\u5747\u4e3api\/n\n    (setq sx 0)\n    (setq i 1)\n    (repeat n\n      (setq xi (cos (\/ (* pi (- i 0.5)) n)))\t\t\t;x \u9879\n      (setq fi (MATH::INT:func xi))\t\t\t\t;x \u9879\u7684\u51fd\u6570\u503c\n      (setq sx (+ sx fi))\n      (setq i  (1+ i))\n    )\n    (setq sx (* wi sx))\t\t\t\t\t\t;\u7edf\u4e00\u4e58\u4ee5\u6743\u503c\n    (if (equal sx s0 eps)\t\t\t\t\t;\u662f\u5426\u6ee1\u8db3\u7cbe\u5ea6\u8981\u6c42\n      (setq flag nil)\t\t\t\t\t\t;\u662f\u5219\u4e0d\u518d\u8fed\u4ee3\n      (setq n (+ n n)\t\t\t\t\t\t;\u5426\u5219\u8fed\u4ee3\u6b21\u6570\u500d\u589e\n\t    s0 sx\t\t\t\t\t\t;\u5b58\u50a8\u79ef\u5206\u503c\n      )\n    )\n  )\n  sx\n)<\/pre>\n
[\/codesyntax]

\u4e0b\u9762\u662f\u4e00\u822c\u51fd\u6570\u7684\u79ef\u5206\u65b9\u6cd5\uff1a
\u516d \u3001\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u9f99\u8d1d\u683c\u79ef\u5206\u6cd5                                                \n;;; Romberg Integration                                         \n;;; \u8f93\u5165: \u51fd\u6570\u540d--foo (\u7528\u7b26\u53f7\u8868\u793a\uff0c\u4e00\u822c\u5f62\u5f0f\u662f (foo x a b c ...) \n;;;       \u53c2\u6570\u8868--args \uff0c\u9664\u53bb\u81ea\u53d8\u91cfx\u7684\u5176\u5b83\u7684\u53c2\u6570\u5217\u8868            \n;;;       \u4e0b\u533a\u95f4--Ra                                            \n;;;       \u4e0a\u533a\u95f4--Rb                                            \n;;;       \u5bb9\u8bb8\u8bef\u5dee--eps                                         \n;;; \u8f93\u51fa: \u6240\u6c42\u51fd\u6570\u5728\u533a\u95f4\u6bb5\u7684\u79ef\u5206                                \n;;;=============================================================\n(defun MATH::INT:Romberg (a b eps \/ EP H I K M N P Q S X Y Y0)\n  (setq h (- b a))\n  (setq y nil)\n  (setq i 0)\n  (repeat 20\n    (setq y (cons (cons i 0.0) y))\n    (setq i (1+ i))\n  )\n  (setq y (reverse y))\n  (setq y0 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq y (cons (cons 0 y0) (cdr y)))\n  (setq\tm  1\n\tn  1\n\tep (1+ eps)\n  )\n  (while (and (>= ep eps) (<= m 19))\n    (setq p 0.0)\n    (setq i 0)\n    (repeat n\n      (setq x (+ a (* (+ i 0.5) h)))\n      (setq p (+ p (MATH::INT:func x)))\n      (setq i (1+ i))\n    )\n    (setq p (\/ (+ (cdar y) (* h p)) 2.0))\n    (setq s 1.0)\n    (setq k 1)\n    (repeat m\n      (setq s (+ s s s s))\n      (setq q (\/ (- (* s p) (cdr (assoc (1- k) y))) (1- s)))\n      (setq y (subst (cons (1- k) p) (assoc (1- k) y) y))\n      (setq p q)\n      (setq k (1+ k))\n    )\n    (setq ep (abs (- q (cdr (assoc (1- m) y)))))\n    (setq m (1+ m))\n    (setq y (subst (cons (1- m) q) (assoc (1- m) y) y))\n    (setq n (+ n n))\n    (setq h (\/ h 2.0))\n  )\n  q\n)<\/pre>\n
[\/codesyntax]
\u4e03 \u8f9b\u666e\u68ee\u79ef\u5206\u6cd5\u3002
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u8f9b\u666e\u68ee\u79ef\u5206\u6cd5                                                \n;;; Simpson Integration                                         \n;;;=============================================================\n(defun MATH::INT:Simpson (a b eps \/ EP H ITER K N P S1 S2 T1 T2 X)\n  (setq n 1)\n  (setq h (- b a))\n  (setq t1 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq s1 t1)\n  (setq ep (1+ eps))\n  (setq iter 0)\n  (while (and (>= ep eps) (< iter 50))\n    (setq p 0.0)\n    (setq k 0)\n    (repeat n\n      (setq x (+ a (* (+ k 0.5) h)))\n      (setq p (+ p (MATH::INT:func x)))\n      (setq k (1+ k))\n    )\n    (setq t2 (\/ (+ t1 (* h p)) 2.))\n    (setq s2 (\/ (- (* 4.0 t2) t1) 3.))\n    (setq ep (abs (- s2 s1)))\n    (setq t1 t2)\n    (setq s1 s2)\n    (setq n (+ n n))\n    (setq h (\/ h 2))\n    (setq iter (1+ iter))\n  )\n  s2\n)\n<\/pre>\n
[\/codesyntax]
\u516b\u3001\u53d8\u6b65\u957f\u68af\u5f62\u6c42\u79ef\u5206\u6cd5 
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u53d8\u6b65\u957f\u68af\u5f62\u6c42\u79ef\u5206\u6cd5                                          \n;;; Trapezoidal Integration 1                                   \n;;;=============================================================\n(defun MATH::INT:Trapezia (a b eps \/ H K N P S T1 T2 X iter)\n  (setq n 1)\n  (setq h (- b a))\n  (setq t1 (* h (+ (MATH::INT:func a) (MATH::INT:func b)) 0.5))\n  (setq p (1+ eps))\n  (setq iter 0)\n  (while (and (>= p eps) (< iter 100))\n    (setq s 0)\n    (setq k 0)\n    (repeat n\n      (setq x (+ a (* (+ k 0.5) h)))\n      (setq s (+ s (MATH::INT:func x)))\n      (setq k (1+ k))\n    )\n    (setq t2 (\/ (+ t1 (* h s)) 2.))\n    (setq p (abs (- t1 t2)))\n    (setq t1 t2)\n    (setq n (+ n n))\n    (setq h (\/ h 2))\n    (setq iter (1+ iter))\n  )\n  t2\n)<\/pre>\n
[\/codesyntax]

\u4e5d\u3001\u6b65\u957f\u79ef\u5206\u6cd5\u3002
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u6b65\u957f\u79ef\u5206\u6cd5                                                  \n;;; Trapezoidal Integration 2                                   \n;;;=============================================================\n(defun MATH::INT:Trapzd (a b n \/ DEL IT SUM TNM X s)\n  (if (= n 1)\n    (setq s (* 0.5 (- b a) (+ (MATH::INT:func a) (MATH::INT:func b))))\n    (progn\n      (setq it 1)\n      (repeat (- n 2)\n\t(setq it (lsh it 1))\n      )\n      (setq tnm it)\n      (setq del (\/ (- b a) tnm))\n      (setq x (+ a (* 0.5 del)))\n      (setq sum 0.0)\n      (repeat it\n\t(setq sum (+ sum (MATH::INT:func x)))\n\t(setq x (+ x del))\n      )\n      (setq s (* 0.5 (+ s (\/ (* (- b a) sum) tnm))))\n    )\n  )\n)<\/pre>\n
[\/codesyntax]

\u5341\u3001\u81ea\u9002\u5e94\u79ef\u5206\u6cd5
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u81ea\u9002\u5e94\u6c42\u79ef\u5206\u6cd5                                              \n;;; Self-adapting Trapezia Integration                          \n;;;=============================================================\n(defun MATH::INT:Atrapezia (a b eps \/ F0 F1 H S T0 TT d)\n  (setq d 1e-4)\n  (setq h (- b a))\n  (setq TT '(0. . 0.))\n  (setq f0 (MATH::INT:func a))\n  (setq f1 (MATH::INT:func b))\n  (setq t0 (* h (+ f0 f1) 0.5))\n  (car (MATH::INT:ppp a b h f0 f1 t0 eps d tt))\n)\n\n(defun MATH::INT:PPP (x0 x1 h f0 f1 t0 eps d tt \/ EPS1 F G P T1 T2 T3 X X2)\n  (setq x (+ x0 (* h 0.5)))\n  (setq f (MATH::INT:func x))\n  (setq t1 (* h (+ f0 f) 0.25))\n  (setq t2 (* h (+ f1 f) 0.25))\n  (setq p (abs (- t0 t1 t2)))\n  (if (or (< p eps) (< (* 0.5 h) d))\n    (cons (+ (car tt) t1 t2) (cdr tt))\n    (progn\n      (setq g (* h 0.5))\n      (setq eps1 (\/ eps 1.4))\n      (setq t3 (MATH::INT:ppp x0 x g f0 f t1 eps1 d tt))\n      (setq t3 (MATH::INT:ppp x x1 g f f1 t2 eps1 d t3))\n    )\n  )\n)<\/pre>\n
[\/codesyntax]
\u4e00\u4e9b\u76f8\u5173\u51fd\u6570\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u529f\u80fd: \u7b2c\u4e00\u7c7b\u692d\u5706\u79ef\u5206                                        \n;;; \u8f93\u5165: Phi \u548c k < 1                                          \n;;; \u8f93\u51fa: \u6240\u6c42\u692d\u5706\u79ef\u5206\u503c                                        \n;;;=============================================================\n(defun Math:Elliptic_Integral_1 (phi kCoff \/ )\n  (defun MATH::INT:func (x \/ s)\n    (setq s (* kCoff (sin x)))\n    (\/ 1.0 (sqrt (* (- 1 s) (1+ s))))\n  )\n  (MATH::INT:Romberg 0 phi 1e-15)\n)\n\n;;;=============================================================\n;;; \u529f\u80fd: \u7b2c\u4e8c\u7c7b\u692d\u5706\u79ef\u5206                                        \n;;; \u8f93\u5165: Phi \u548c k < 1                                          \n;;; \u8f93\u51fa: \u6240\u6c42\u692d\u5706\u79ef\u5206\u503c                                        \n;;;=============================================================\n(defun Math:Elliptic_Integral_2 (phi kCoff \/ )\n  (defun MATH::INT:func (x \/)\n    (setq x (* kCoff (sin x)))\n    (sqrt (* (- 1 x) (1+ x)))\n  )\n  (MATH::INT:Romberg 0 phi 1e-15)\n)\n\n;;;=============================================================\n;;; \u9636\u4e58                                                        \n;;;=============================================================\n(defun Math::frac (n)\n  (if (= n 0)\n    1.\n    (if\t(= n 1)\n      1.\n      (* n (Math::frac (1- n)))\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u4f3d\u739b\u51fd\u6570                                                    \n;;;=============================================================\n(defun Math::gammln (xx \/ x y tmp ser j cof)\n  (setq\tcof '(76.18009172947146\t\t -86.50532032941677\n\t      24.01409824083091\t\t -1.231739572450155\n\t      0.1208650973866179e-2\t -0.5395239384953e-5\n\t     )\n  )\n  (setq x xx)\n  (setq y x)\n  (setq tmp (+ x 5.5))\n  (setq tmp (- tmp (* (+ 0.5 x) (log tmp))))\n  (setq ser 1.000000000190015)\n  (setq j 0)\n  (repeat 6\n    (setq y (1+ y))\n    (setq ser (+ ser (\/ (nth j cof) y)))\n    (setq j (1+ j))\n  )\n  (- (log (\/ (* 2.5066282746310005 ser) x)) tmp)\n)\n\n;;;=============================================================\n;;; \u6b21\u51fd\u6570\u7528\u4e8e\u6bd4\u8f83\u6392\u5e8f                                          \n;;;=============================================================\n(defun MATH::INT:funcSort (e1 e2)\n  (< (car e1) (car e2))\n)\n\n;;;=============================================================\n;;; \u7ec4\u5408\u4e24\u4e2a\u7cfb\u6570                                                \n;;;=============================================================\n(defun MATH::INT:Bind (xi wi)\n  (setq xi (vl-sort xi 'MATH::INT:funcSort))\n  (setq wi (vl-sort wi 'MATH::INT:funcSort))\n  (setq xi (mapcar 'cdr xi))\n  (setq wi (mapcar 'cdr wi))\n  (mapcar 'cons xi wi)\n)<\/pre>\n
[\/codesyntax]


\u4e0b\u9762\u662f\u7528\u5bf9\u8bdd\u6846\u521b\u5efa\u7684\u6c42\u79ef\u5206\u6cd5\u7684\u4f8b\u7a0b\uff1a

[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u7528\u5404\u79cd\u65b9\u6cd5\u6c42\u79ef\u5206\u7684\u7a0b\u5e8f                                      \n;;;=============================================================\n(defun C:Quadrature (\/ ID OK DCL_FILE)\n  (setq id (load_dialog (setq Dcl_File (MATH::INT:Write_Dcl))))\t;\u4ece\u5bf9\u8bdd\u6846\u4e2d\u5f97\u5230\u8868\u8fbe\u5f0f\n  (vl-file-delete Dcl_File)\t\t\t\t\t;\u5220\u9664\u4e34\u65f6\u5bf9\u8bdd\u6846\u6587\u4ef6\n  (setq ok 2)\n  (if (new_dialog \"dcl_Integration\" id)\n    (progn\n      (VL-CATCH-ALL-APPLY 'MATH::INT:GetSettings)\t\t;\u8bfb\u53d6\u9ed8\u8ba4\u6570\u636e\n      (action_tile \"help\" \"(MATH::INT:Help 1)\")\t\t\t;\u5e2e\u52a9\n      (foreach k '(0 1 2 3 4 5 6 7 8)\n\t(setq k (strcat \"K\" (itoa k)))\n        (action_tile k \"(MATH::INT:OnBtn $key)\")                ;\u6309\u94ae\u52a8\u4f5c\uff0c\u5bf9\u5e94\u76f8\u5bf9\u7684\u79ef\u5206\u65b9\u6cd5\n      )\t\t   \n      (setq ok (start_dialog))\n    )\n  )\n  (unload_dialog ID)\n  (princ)\n)\n\n(defun C:JF (\/)\n  (VL-CATCH-ALL-APPLY 'C:Quadrature)\n  (princ)\n)\n\n;;;=============================================================\n;;; \u4ece\u73af\u5883\u53d8\u91cf\u8bfb\u53d6\u4e0a\u6b21\u6570\u636e                                      \n;;;=============================================================\n(defun MATH::INT:GetSettings (\/ data)\n  (if (setq Data (getenv \"Intergration\"))\n    (foreach k (read data)\n      (set_tile (car k) (cdr k))\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u68c0\u67e5\u5bf9\u8bdd\u6846\u8f93\u5165                                              \n;;;=============================================================\n(defun MATH::INT:CheckInput (symS symA symB symN \/ e f)\n  (setq e (exp 1))\n  (set symS (get_tile \"F\"))\n  (set symA (MATH::INT:MyRead (get_tile \"A\")))\n  (set symB (MATH::INT:MyRead (get_tile \"B\")))\n  (set symN (MATH::INT:MyRead (get_tile \"N\")))\n  (setq f (CAL:Expr2Func (eval s) 'MATH::INT:func '(x)))\n  (apply 'and (mapcar 'eval '(F symS symA symB symN)))   \n)\n\n\n;;;=============================================================\n;;; \u6309\u94ae\u52a8\u4f5c\uff0c\u5bf9\u5e94\u76f8\u5e94\u7684\u51fd\u6570\u6c42\u79ef\u5206                              \n;;;=============================================================\n(defun MATH::INT:OnBtn (key \/ DATA s N a b OldZIN m EPS RET tm0 map msg foo tmp idx)\n  (setq m (VL-CATCH-ALL-APPLY 'MATH::INT:CheckInput '(s a b n)))\n  (if (or (vl-catch-all-error-p m) (not m) (equal a b 1e-8))\n    (if (vl-catch-all-error-p m) \n      (set_tile \"info\" (vl-catch-all-error-message m))\n      (set_tile \"info\" \"\u65e0\u6548\u8f93\u5165!\")\n    )\n    (progn\n      ;;\u5982\u679c\u7cbe\u5ea6\u8fc7\u9ad8\uff0c\u8bbe\u7f6e\u4e3a15\u4f4d\u7684\u7cbe\u5ea6\n      (if (> n 20)\n\t(setq n 15)\n\t(setq n (fix (abs n)))\n      )\n      ;;\u5982\u679c\u4e0a\u533a\u95f4\u5c0f\u4e8e\u4e0b\u533a\u95f4\uff0c\u5219\u4ea4\u6362\u533a\u95f4\n      (if (< b a)\n\t(setq tmp a a b b tmp)\n      )\n      ;;\u8bb0\u4f4f\u5bf9\u8bdd\u6846\u8f93\u5165\uff0c\u7528\u4e8e\u4e0b\u6b21\n      (setq OldZIN (getvar \"DIMZIN\"))\n      (setvar \"DIMZIN\" 8)\n      (set_tile \"N\" (itoa n))\n      (set_tile \"A\" (rtos a 2 20))\n      (set_tile \"B\" (rtos b 2 20))\n      (setq data (list (cons \"F\" s)\n\t\t       (cons \"N\" (itoa n))\n\t\t       (cons \"A\" (rtos a 2 20))\n\t\t       (cons \"B\" (rtos b 2 20))\n\t\t )\n      )\n      (setvar \"DIMZIN\" OldZIN)\n      (setenv \"Intergration\" (VL-PRIN1-TO-STRING data))\n      ;;\u5f00\u59cb\u8ba1\u7b97\u79ef\u5206\n      (setq eps (expt 0.1 n))\n      (setq tm0 (getvar \"TDUSRTIMER\"))\n      (setq map (MATH::INT:GetMethods))\t\t\t\t;\u79ef\u5206\u8ba1\u7b97\u65b9\u6cd5\u96c6\n      (setq idx (atoi (substr key 2)))\t\t\t\t\n      (setq foo (nth idx map))\t\t\t\t\t;\u83b7\u53d6\u8ba1\u7b97\u79ef\u5206\u7684\u51fd\u6570\n      (setq ret (VL-CATCH-ALL-APPLY foo (list a b eps)))        ;\u83b7\u53d6\u79ef\u5206\u503c\n      (if (vl-catch-all-error-p ret)\n\t(set_tile \"info\" (vl-catch-all-error-message ret))\t;\u6c42\u89e3\u8fc7\u7a0b\u53d1\u751f\u4e86\u9519\u8bef\n        (if (null ret)\n\t  (set_tile \"info\" \"\u53d1\u751f\u4e86\u9519\u8bef\uff0c\u6c42\u503c\u7ed3\u679c\u4e3a\u7a7a!\")\t\n\t  (progn\n\t    ;(MATH::INT:Bench 100 a b eps)\n            (setq ret (rtos ret 2 20))\n            (setq msg (get_attr key \"label\"))\n            (setq msg (strcat msg \"\u6c42\u7684\u7ed3\u679c\u4e3a:\" ret))\n            (set_tile (strcat \"R\" (itoa idx)) ret)             \t;\u663e\u793a\u6c42\u89e3\u7ed3\u679c\n            (set_tile \"info\" msg)\t\t\t\t;\u663e\u793a\u6c42\u89e3\u7ed3\u679c\n            (princ (strcat \"\\n\" msg))\t\t\t\t;\u6253\u5370\u6c42\u89e3\u7ed3\u679c\n            (princ \"\\n\u8d39\u65f6:\")\n            (princ (* (- (getvar \"TDUSRTIMER\") tm0) 86400))\n            (princ \"\u79d2.\")\n\t  )\n        )\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u5404\u79cd\u79ef\u5206\u6d4b\u901f                                                \n;;;=============================================================\n(defun MATH::INT:Bench (n a b eps)\n  (UTI:BENCH\n    n\n    (list\n      (list 'MATH::INT:Romberg a b eps)\n      (list 'MATH::INT:Gauss-Legendre a b eps)\n      (list 'MATH::INT:Simpson a b eps)\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u79ef\u5206\u65b9\u6cd5\u96c6                                                  \n;;;=============================================================\n(defun MATH::INT:GetMethods ()\n  '(MATH::INT:Romberg\n    MATH::INT:Simpson\n    MATH::INT:Atrapezia\n    MATH::INT:Trapezia\n    MATH::INT:Gauss-Legendre\n    Math::INT:Gauss-Chebyshev\n    Math::INT:Gauss-Laguerre\n    Math::INT:Gauss-Hermite\n    MATH::INT:Gauss-Jacobi\n   )\n)\n\n;;;=============================================================\n;;; \u8868\u8fbe\u5f0f\u6c42\u503c\uff0c\u4e5f\u53ef\u4ee5\u7528cal\u51fd\u6570                                 \n;;;=============================================================\n(defun MATH::INT:MyRead (str \/ e)\n  (setq e (exp 1))\n  (CAL:Expr2Value str)\n)\n\n;;;=============================================================\n;;; \u5e2e\u52a9\u548c\u8bf4\u660e: help and instruction                            \n;;;=============================================================\n(defun MATH::INT:Help (n)\n  (if (= n 1)\n    (if\t(= \"CHS\" (getvar \"Locale\"))\n      (alert\n\t\"\u51fd\u6570\u5f0f\u53ea\u63a5\u53d7\u7b26\u53f7x\u4e3a\u53d8\u91cf,\u4e0d\u89c4\u8303\u5f88\u53ef\u80fd\u51fa\u9519!\n\t\\n\u51fd\u6570\u53ef\u4ee5LISP\u5185\u7f6e\u7684\u6570\u5b66\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u81ea\u5b9a\u4e49\u51fd\u6570!\n\t\\n\u6307\u6570\u7528^\u8868\u793a\uff0c+-*\/\u8868\u793a\u52a0\u51cf\u4e58\u9664\uff0c\u4e58\u53f7\u4e0d\u80fd\u7701\u7565\u3002\n\t\\n\u7a0b\u5e8f\u80fd\u91c7\u7528\u591a\u79cd\u65b9\u6cd5\u6c42\u79ef,\u4e00\u822c\u6765\u8bf4\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\u6700\u5feb\u3002\n\t\\n\u6709\u4ec0\u4e48\u95ee\u9898email: highflybird@qq.com\n\t\\n\u4f5c\u8005: highflybird \u65e5\u671f2019.07\"\n      )\n      (alert\n\t\"Standard expression only accepts \\\"x\\\" as a variale!\n\t\\nThe fastest is Romberg Integration,the slowest is Trapezoidal rule(Be careful!).\n\t\\nRecommendation:Don't set a high precision at first,promote it step by step.\n\t\\nEspically for the Trapezoidal rule, It won't work well on some circumstances.\n\t\\nIt's an Open Source Software. Thanks for your advice or bug reports.\n\t\\nAuthor: highflybird  Email: highflybird@qq.com  Date:2019.07.\"\n      )\n    )\n    (set_tile \"info\" \"\u8868\u8fbe\u5f0f\u975e\u6cd5\u6216\u8005\u7a7a\u8f93\u5165.\")\n  )\n)\n<\/pre>\n
[\/codesyntax]
\u5bf9\u8bdd\u6846\u7684\u5236\u4f5c\uff1a
[codesyntax lang=\"cadlisp\"]<\/pre>\n
;;;=============================================================\n;;; \u8f93\u5165\u5bf9\u8bdd\u6846                                                  \n;;;=============================================================\n(defun UTI:Inputbox (\/ str wcs ret)\n  (setq\tstr \"Function GetNumbers()\n  \t     GetNumbers=inputbox(\\\"\u8bf7\u8f93\u5165\u4e24\u4e2a\u53c2\u6570,\u4e2d\u95f4\u7528\u7a7a\u683c\u9694\u5f00:\\\",\\\"\u8f93\u5165\u6846\\\")\n             End Function\"\n  )\n  (if\n    (or\n      (setq wcs (vlax-create-object \"Aec32BitAppServer.AecScriptControl.1\"))\n      (setq wcs (vlax-create-object \"ScriptControl\"))\n    )\n    (progn \n      (vlax-put-property wcs \"language\" \"VBScript\")\n      (vlax-invoke wcs 'addcode str)\n      (if (setq ret (vlax-invoke wcs 'run \"GetNumbers\"))\n\t(setq ret (strcat \"(\" ret \")\")\n\t      ret (read ret)\n\t)\n      )\n      (vlax-release-object wcs)\n      (if\n\t(and\n\t  (= 2 (length ret))\n\t  (or (= 'INT (type (car ret))) (= 'REAL (type (car ret))))\n\t  (or (= 'INT (type (cadr ret))) (= 'REAL (type (cadr ret))))\n\t)\n\tret\n      )\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u5199\u5bf9\u8bdd\u6846\u5230\u6587\u4ef6\u7528\u4e8e\u7a0b\u5e8f                                      \n;;;=============================================================\n(defun MATH::INT:Write_Dcl (\/ Dcl_File file str)\n  (setq Dcl_File (vl-filename-mktemp nil nil \".Dcl\"))\n  (setq file (open Dcl_File \"w\"))\n  (princ\n    \"dcl_Integration : dialog {\n\tlabel = \\\"\u6570\u503c\u79ef\u5206LISP\u7248  v1.2\\\";\n\t: boxed_column {\n          width = 60;\n\t  fixed_width = true;\n\t  : edit_box {\n\t    key=\\\"F\\\";\n\t    label= \\\"\u51fd\u6570:\\\";\n\t  }\n\t  : row {\n\t    : edit_box {\n\t      key=\\\"A\\\";\n\t      label= \\\"\u4e0b\u9650:\\\";\n\t    }\n\t    : edit_box {\n\t      key=\\\"B\\\";\n\t      label= \\\"\u4e0a\u9650:\\\";\n\t    }      \n\t    : edit_box {\n\t      key=\\\"N\\\";\n\t      label = \\\"\u7cbe\u786e\u4f4d\u6570:\\\";\n\t      value = 8;\n\t      edit_width = 2;\n\t      fixed_width = true;\n\t    }\n\t  }\n\t  spacer_1;\n\t}\n\t: row {\n\t  : boxed_column {\n\t    label = \\\"\u8ba1\u7b97\u65b9\u6cd5:\\\";\n\t    : button {\n\t      key = \\\"K0\\\";\n\t      label = \\\"\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K1\\\";\n\t      label = \\\"\u8f9b\u666e\u68ee\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K2\\\";\n\t      label = \\\"\u81ea\u9002\u5e94\u79ef\u5206\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K3\\\";\n\t      label = \\\"\u53d8\u6b65\u957f\u68af\u5f62\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K4\\\";\n\t      label = \\\"\u9ad8\u65af-\u52d2\u8ba9\u5fb7\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K5\\\";\n\t      label = \\\"\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K6\\\";\n\t      label = \\\"\u9ad8\u65af-\u62c9\u76d6\u5c14\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K7\\\";\n\t      label = \\\"\u9ad8\u65af-\u57c3\u5c14\u7c73\u7279\u6cd5\\\";\n\t    }\n\t    : button {\n\t      key = \\\"K8\\\";\n\t      label = \\\"\u9ad8\u65af-\u96c5\u514b\u6bd4\u6cd5\\\";\n\t    }\n\t    spacer;\n\t  }\n\t  : boxed_column {\n\t    width = 32;\n\t    fixed_width = true;\n\t    label = \\\"\u8ba1\u7b97\u7ed3\u679c:\\\";\n\t    : text {\n\t      key = \\\"R0\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R1\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R2\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R3\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R4\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R5\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R6\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R7\\\";\n\t    }\n\t    : text {\n\t      key = \\\"R8\\\";\n\t    }\n\t    spacer;\n\t  }\n\t}\n\tok_cancel_help;\n\t\/\/ok_cancel_help_errtile;\n\t: text {\n\t  key = \\\"info\\\";\n\t  label = \\\"Copyright \\\\u+00A9 2007-2019 Highflybird. All rights reserved.\\\";\n\t  width = 20;\n\t}\n    }  \"\n    file\n  )\n  (close file)\n  Dcl_File\n)\n\n;;;=============================================================\n;;; \u4e0b\u9762\u7684\u5c31\u4e0d\u7528\u4ecb\u7ecd\u4e86                                          \n;;;=============================================================\n(vl-load-com)\n(if (= \"CHS\" (getvar \"Locale\"))\n  (prompt \"\u8f93\u5165\u547d\u4ee4: JF\")\n  (prompt \"Please enter: Quadrature\")\n)\n(c:JF)\n(princ)<\/pre>\n
[\/codesyntax]

<\/pre>\n\n\n
\u4e00\u4e9b\u5e94\u7528\uff0c\u5982\u4e0b\u9762\u7f51\u53cb\u7684\u63d0\u95ee\u5e76\u89e3\u7b54\uff1a<\/p>\n\n\n\n
http:\/\/bbs.mjtd.com\/forum.php?mod=viewthread&tid=179809&extra=&highlight=%BB%FD%B7%D6&page=1<\/a><\/p>\n\n\n\n
\n\n\u692d\u7403\u4f53\u7403\u7f3a\u7684\u8868\u9762\u79ef\u8ba1\u7b97
\u7528\u79ef\u5206<\/strong>\u516c\u5f0f\u7528LISP\u7a0b\u5e8f\u8868\u8fbe\u5e76\u8ba1\u7b97\u51fa\u6765
\u8fd9\u662f\u4e00\u4e2a\u7f51\u4e0a\u7684
\u6709\u7ed3\u679c\u548c\u516c\u5f0f\n\n<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\u7528\u4e0a\u9762\u7684\u4ecb\u7ecd\u7684\u4e00\u4e9b\u51fd\u6570\u53ef\u4ee5\u5f97\u51fa\u5176\u9762\u79ef\u548c\u4f53\u79ef\uff1a<\/p>\n\n\n
[codesyntax lang=”cadlisp”]<\/p>\n
;;;=============================================================\n;;; \u529f\u80fd: \u4e3b\u7a0b\u5e8f\uff0c\u83b7\u53d6\u692d\u7403\u51a0\u7684\u9762\u79ef                              \n;;; \u8f93\u5165: \u65e0\u3002                                                  \n;;; \u8f93\u51fa: \u65e0\u3002                                                  \n;;;=============================================================\n(defun c:test (\/ a b c d h s0 s1)\n  (initget 15)\n  (setq a (getdist \"\\n\u8f93\u5165\u692d\u7403X\u534a\u8f74\u957f: \"))\t\t\t\n  (initget 15)\n  (setq b (getdist \"\\n\u8f93\u5165\u692d\u7403y\u534a\u8f74\u957f: \"))\n  (initget 15)\n  (setq c (getdist \"\\n\u8f93\u5165\u692d\u7403z\u534a\u8f74\u957f: \"))\n  (initget 15)\n  (setq d (getdist \"\\n\u8f93\u5165\u692d\u7403\u51a0\u7684\u9ad8\u5ea6: \"))\t\t\t\n  (if (<= d (+ c c))\t\t\t\t\t\t;\u6240\u6c42\u8303\u56f4\u4e3a\u5c0f\u4e8e\u692d\u7403Z\u8f74\u957f\n    (progn\n      (setq h (\/ (- c d) c))\n      (setq s0 (ELL:GetSurfaceArea a b c 1))\t\t\t;\u7528\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\u8ba1\u7b97\u5176\u534a\u4e2a\u692d\u7403\u9762\u79ef\n      (setq s2 (ELL:GetSurfaceArea a b c 1))\t\t\t;\u7528\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206\u6cd5\u8ba1\u7b97\u5176\u534a\u4e2a\u692d\u7403\u9762\u79ef\n      (if (zerop h)\t\t\t\t\t\t;\u4e3a\u4e86\u9632\u6b62\u88ab\u96f6\u9664\n\t(setq s1 0 s2 0)\n\t(setq s1 (ELL:GetSurfaceArea a b c h) \t\t\t;\u7528\u9f99\u8d1d\u683c\u79ef\u5206\u6cd5\u8ba1\u7b97\u692d\u7403\u5e26\u9762\u79ef\n\t      s3 (ELL:GetSurfaceArea3 a b c h)\t\t\t;\u7528\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206\u6cd5\u8ba1\u7b97\u692d\u7403\u5e26\u9762\u79ef\n\t)\n      )\n      (princ \"\\n\u7528\u9f99\u8d1d\u683c\u6cd5\u5f97\u5230\u6240\u6c42\u8868\u9762\u79ef\u662f: \")\n      (princ (rtos (- s0 s1) 2 20))\t\t\t\t;\u4e24\u4e2a\u9762\u79ef\u76f8\u51cf\uff0c\u4fbf\u662f\u5176\u692d\u7403\u51a0\u9762\u79ef\n\n      (princ \"\\n\u7528\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206\u6cd5\u5f97\u5230\u6240\u6c42\u8868\u9762\u79ef\u662f: \")\n      (princ (rtos (- s2 s3) 2 20))\t\t\t\t;\u4e24\u4e2a\u9762\u79ef\u76f8\u51cf\uff0c\u4fbf\u662f\u5176\u692d\u7403\u51a0\u9762\u79ef\n\n      (princ \"\\n\u7528\u9f99\u8d1d\u683c\u6cd5\u6240\u6c42\u7684\u692d\u7403\u7684\u8868\u9762\u79ef\u662f: \")\n      (princ (rtos (+ s0 s0) 2 20))\n\n      (princ \"\\n\u4f30\u7b97\u7ed3\u679c\u662f: \")\n      (princ (rtos (ELL:GetSurfaceArea2 a b c) 2 20))\n      \n      (princ \"\\n\u53e6\u5916\u7684\u7ed3\u679c\u662f: \")\n      (princ (rtos (ELL:GetSurfaceArea1 a b c) 2 20))\n\n      (princ \"\\n\u9ad8\u65af-\u5207\u6bd4\u96ea\u592b\u79ef\u5206\u7684\u7ed3\u679c\u662f: \")\n      (princ (rtos (+ s2 s2) 2 20))\n    )\n  )\n  (princ)\n)\n\n;;;=============================================================\n;;; \u529f\u80fd: \u83b7\u53d6\u692d\u7403\u5e26\u7684\u9762\u79ef                                      \n;;; \u8f93\u5165: \u692d\u7403\u7684\u4e09\u4e2a\u65b9\u5411\u7684\u8f74\u5f84\uff0c\u548c\u692d\u7403\u5e26\u7684\u9ad8\u5ea6                  \n;;; \u8f93\u51fa: \u692d\u7403\u5e26\u7684\u9762\u79ef                                          \n;;;=============================================================\n(defun ELL:GetSurfaceArea (a b c h \/ func)\n  (defun Func (x a b c h \/ F G K S Y Z)\n    (setq k (* a a))\n    (setq s (sin x))\n    (setq g (- 1 (* (- 1 (\/ k b b)) s s)))\n    (setq z (1- (\/ k c c g)))\n    (setq y (1+ (* z h h)))\n    (if\t(zerop z)\n      (* (sqrt g) (1+ (sqrt y)))\n      (if (< z 0)\n\t(setq z\t(* h (sqrt (- z)))\n\t      f\t(* (sqrt g) (+ (sqrt y) (\/ (asin z) z)))\n\t)\n\t(setq z\t(* h (sqrt z))\n\t      f\t(* (sqrt g) (+ (sqrt y) (\/ (asinh z) z)))\n\t)\n      )\n    )\n  )\n  (* 2 b c h (Math:Romberg 'func (list a b c h) 0 (* 0.5 pi) 1e-15))\n)\n\n;;;=============================================================\n;;; \u529f\u80fd: \u7528\u692d\u5706\u51fd\u6570\u8ba1\u7b97\u692d\u7403\u8868\u9762\u79ef                              \n;;;=============================================================\n(defun ELL:GetSurfaceArea1 (a b c \/ ll aa bb cc ac e1 e2 ph k f1 f2 s)\n  (if (and (equal a b 1e-8) (equal b c 1e-8))\n    (* 4 pi a)\n    (setq ll (vl-sort (list a b c) '>)\n\t  a  (car ll)\n\t  b  (cadr ll)\n\t  c  (caddr ll)\n\t  aa (* a a)\n\t  bb (* b b)\n\t  cc (* c c)\n\t  ac (sqrt (- aa cc))\n\t  e1 (\/ ac a)\n\t  e2 (\/ (sqrt (- bb cc)) b)\n\t  ph (asin e1)\n\t  k  (\/ e2 e1)\n\t  f1 (Math:Elliptic_Integral_1 ph k)\n\t  f2 (Math:Elliptic_Integral_2 ph k)\n\t  s  (* 2 pi (+ cc (\/ (* b cc f1) ac) (* b ac f2)))\n    )\n  )\n)\n\n;;;=============================================================\n;;; \u529f\u80fd: \u4f30\u7b97\u692d\u7403\u8868\u9762\u79ef                                        \n;;;=============================================================\n(defun ELL:GetSurfaceArea2 (a b c)\n  (* 4\n     pi\n     (expt\n       (\/ (+ (expt (* a b) 1.6075)\n\t     (expt (* b c) 1.6075)\n\t     (expt (* c a) 1.6075)\n\t  )\n\t  3\n       )\n       (\/ 1 1.6075)\n     )\n  )\n)\n\n\n\n;;;=============================================================\n;;; \u51e0\u4e2a\u76f8\u5173\u6570\u5b66\u51fd\u6570                                            \n;;;=============================================================\n(defun asin (x) (atan x (sqrt (- 1 (* x x))))) \t\t\t;\u53cd\u6b63\u5f26\u51fd\u6570\n(defun asinh (x) (log (+ x (sqrt (1+ (* x x))))))\t\t;\u53cd\u53cc\u66f2\u6b63\u5f26\u51fd\u6570=log(x+sqrt(x*x+1))\n\n(vl-load-com)\n(princ \"\\n\u8fd0\u884c\u547d\u4ee4\u662f:Test\")\n(princ)<\/pre>\n
[\/codesyntax]<\/p>\n\n\n
\n
\u81f3\u4e8e\u4e00\u4e9b\u79ef\u5206\u51fd\u6570\u6709\u4ec0\u4e48\u533a\u522b\uff0c\u6211\u5c31\u4e0d\u5728\u8fd9\u91cc\u4e00\u4e00\u4ecb\u7ecd\u4e86\u3002<\/p>\n<\/div><\/div>\n\n\n\n
\n
\u5e0c\u671b\u8fd9\u4e9b\u4ee3\u7801\u5bf9\u4f60\u6709\u5e2e\u52a9\u3002\u5f15\u7528\u8bf7\u6ce8\u660e\u6765\u6e90\u3002<\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"
\u7528LISP\u7f16\u5199\u4e86\u4e00\u4e2a\u6c42\u79ef\u5206\u7684\u7a0b\u5e8f\uff1a \u91cc\u9762\u91c7\u7528\u4e86\u5404\u79cd\u65b9\u6cd5\u6c42\u79ef\u5206\u548c\u5404\u79cd\u7c7b\u578b\u7684\u79ef\u5206\u3002\u4e0b\u9762\u6211\u628a\u5404\u79cd\u65b9\u6cd5\u7684\u6e90\u7801\u8d34\u51fa\u3002<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-4815","post","type-post","status-publish","format-standard","hentry","category-programming"],"_links":{"self":[{"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/4815","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4815"}],"version-history":[{"count":0,"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/4815\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4815"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4815"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.highflybird.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}