Typically when my MATLAB is busy I take the opportunity to get more coffee.
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my matlab is busy!!
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if i turn on the code
matlab is busy infinitly(may be not).
[R2,x]=solve('((1-R2)/(1-R1)*R1-(c/b*R2))*x^2+(1-R2)/(1-R1)-c/b=0','((2-R1-R2)/(1-R1))*a*x^2+(b+c)*(1-R2)*x+a*(((1-R2)/(1-R1)+R1)+R2)=0','R2,x')
my equation is too complex? give me a tip to slove this equation
3 件のコメント
Jan
2011 年 8 月 22 日
Typically when my MATLAB is busy it takes the opportunity to get more coffee.
@Yunho: How long is "infinitely" at the moment?
回答 (3 件)
Walter Roberson
2011 年 8 月 22 日
It only takes a short time in Maple. There are four solutions. You will have to excuse me for not posting them in full: it takes to long to format them.
The following MATLAB code fragment will calculate R2 for the first of the solutions:
t1 = (a ^ 2);
t2 = (b * t1);
t4 = 2 - R1;
t6 = R1 * b;
t7 = c * t4 + t6;
t9 = c ^ 2;
t10 = R1 * t9;
t11 = b * c;
t13 = -t10 + t11 + t9 - R1 * t11;
t14 = t13 ^ 2;
t15 = a * R1;
t16 = c * t15;
t17 = b * t15;
t19 = 2 * a * c;
t20 = -t16 + t17 + t19;
t21 = t20 ^ 2;
t22 = 1 / t21;
t23 = t22 * t14;
t25 = R1 ^ 2;
t26 = t25 * a;
t30 = b * t26 + t17 - c * t26 + t16 + a * b + t19;
t31 = 1 / t20;
t32 = t31 * t30;
t33 = t25 * R1;
t38 = b ^ 2;
t41 = b * t9;
t49 = t9 * c;
t51 = t25 * t49;
t55 = t38 * b;
t58 = 3 * b * t33 * t9 - 3 * t38 * c * t33 - 12 * t25 * t41 + 6 * c * t38 * t25 + 12 * t9 * t6 - t49 * t33 + 6 * t51 - 12 * t49 * R1 + t55 * t33 + 8 * t49;
t60 = 1 / a;
t62 = 8 ^ (0.1e1 / 0.3e1);
t63 = t7 ^ 2;
t64 = t63 ^ 2;
t66 = -1 + R1;
t67 = t66 ^ 2;
t68 = t67 ^ 2;
t69 = t68 * t67;
t70 = t9 ^ 2;
t71 = t70 ^ 2;
t79 = t25 ^ 2;
t80 = t1 * t79;
t81 = t1 * t33;
t84 = 60 * t38;
t91 = 32 * t1;
t100 = 40 * t38;
t112 = t67 * t66;
t113 = t1 ^ 2;
t114 = t38 * t1;
t117 = t79 * R1;
t125 = t38 ^ 2;
t126 = 0.15e2 / 0.22e2 * t125;
t131 = 0.45e2 / 0.22e2 * t125;
t135 = 4 * t113;
t145 = t79 * t25;
t155 = 3 * t125;
t160 = 9 * t125;
t165 = 6 * t113;
t176 = t79 ^ 2;
t177 = t113 * t1;
t179 = t38 * t113;
t183 = t79 * t33;
t186 = t125 * t1;
t197 = t125 * t38;
t198 = t197 / 0.4e1;
t204 = 16 * t177;
t220 = (t177 * t176) + t183 * (t179 / 0.2e1 - (4 * t177)) + t145 * ((5 * t177) + t186 / 0.16e2 + 0.71e2 / 0.4e1 * t179) + t117 * (-(10 * t177) - 0.21e2 / 0.4e1 * t186 - 0.193e3 / 0.4e1 * t179) + t79 * (0.19e2 / 0.8e1 * t186 + t198 + (15 * t179) + (24 * t177)) + t33 * (-0.171e3 / 0.4e1 * t179 - t197 - t204 + 0.51e2 / 0.4e1 * t186) + t25 * (0.3e1 / 0.2e1 * t197 + t204 - 0.367e3 / 0.16e2 * t186 + t179 / 0.4e1) + R1 * (-t197 + (24 * t186) + 0.155e3 / 0.2e1 * t179 - (32 * t177)) + t198 - (11 * t186) - (22 * t179);
t228 = 0.7e1 / 0.4e1 * t113;
t232 = 2 * t113;
t245 = 0.7e1 / 0.16e2 * t125;
t261 = (t113 * t176) + t183 * (-t114 / 0.4e1 - t228) + t145 * (0.7e1 / 0.8e1 * t114 - t232 + t125 / 0.32e2) + t117 * (-t228 + 0.55e2 / 0.8e1 * t114 + 0.7e1 / 0.4e1 * t125) + t79 * (0.47e2 / 0.8e1 * t114 + 0.13e2 / 0.2e1 * t113 - 0.23e2 / 0.8e1 * t125) + t33 * (t165 - t245 - 0.75e2 / 0.4e1 * t114) + t25 * (t135 + 0.91e2 / 0.32e2 * t125 + 0.167e3 / 0.16e2 * t114) + R1 * (-0.29e2 / 0.16e2 * t125 - (12 * t113) + 0.15e2 / 0.16e2 * t114) + t125 / 0.2e1 + (8 * t113) - 0.7e1 / 0.2e1 * t114;
t263 = t70 * c;
t268 = t176 * R1;
t302 = 0.64e2 / 0.5e1 * t113;
t308 = (t113 * t268) - (t38 * t1 * t176) / 0.5e1 + t183 * (-0.18e2 / 0.5e1 * t114 - t125 / 0.80e2 - 0.26e2 / 0.5e1 * t113) + t145 * (-t245 + (9 * t114) - 0.4e1 / 0.5e1 * t113) + t117 * (0.27e2 / 0.5e1 * t113 + 0.57e2 / 0.40e2 * t125 - 0.41e2 / 0.10e2 * t114) + t79 * (t232 - 0.59e2 / 0.40e2 * t125 + 0.7e1 / 0.5e1 * t114) + t33 * (0.43e2 / 0.80e2 * t125 - 0.12e2 / 0.5e1 * t113 - 0.111e3 / 0.20e2 * t114) + t25 * (-0.43e2 / 0.10e2 * t114 + 0.8e1 / 0.5e1 * t113 - 0.3e1 / 0.16e2 * t125) + R1 * (t125 / 0.4e1 + t302 + 0.47e2 / 0.5e1 * t114) - 0.41e2 / 0.20e2 * t114 - t125 / 0.10e2 - t302;
t314 = t38 / 0.20e2;
t318 = 0.9e1 / 0.40e2 * t38;
t321 = 0.3e1 / 0.2e1 * t38;
t329 = 0.9e1 / 0.2e1 * t38;
t338 = 0.34e2 / 0.5e1 * t1;
t341 = 0.28e2 / 0.5e1 * t1;
t386 = (t1 * t176 * t25) - (t38 * t268) / 0.10e2 + t176 * (-0.23e2 / 0.20e2 * t38 - t338) + t183 * (0.49e2 / 0.20e2 * t38 - t341) + t145 * (0.61e2 / 0.5e1 * t1 - 0.9e1 / 0.20e2 * t38) + t117 * ((18 * t1) - 0.13e2 / 0.10e2 * t38) + t79 * (-0.32e2 / 0.5e1 * t1 + 0.17e2 / 0.10e2 * t38) + t33 * (-0.9e1 / 0.5e1 * t38 - 0.114e3 / 0.5e1 * t1) + t25 * (t314 + 0.11e2 / 0.5e1 * t1) + R1 * (0.13e2 / 0.20e2 * t38 + 0.58e2 / 0.5e1 * t1) - 0.16e2 / 0.5e1 * t1 - t314;
t398 = -1 + R1 + t25;
t399 = t398 ^ 2;
t405 = R1 + 1;
t407 = t399 ^ 2;
t411 = (12 * t71 * t70 * t69) + (72 * t71 * t49 * t69 * b) - (3 * t71 * t9 * (t80 - 2 * t81 + t25 * (-87 * t1 - t84) + R1 * (136 * t1 + 120 * t38) - t84 - t91) * t68) - (6 * t71 * c * (t80 - 22 * t81 + t25 * (-145 * t1 - t100) + R1 * (80 * t38 + 254 * t1) - t91 - t100) * b * t68) - 0.264e3 * t71 * (t117 * (t113 - t114 / 0.88e2) + t79 * (-0.32e2 / 0.11e2 * t113 - 0.117e3 / 0.88e2 * t114) + t33 * (-0.175e3 / 0.88e2 * t114 - 0.13e2 / 0.11e2 * t113 - t126) + t25 * (0.961e3 / 0.88e2 * t114 + 0.86e2 / 0.11e2 * t113 + t131) + R1 * (-0.153e3 / 0.22e2 * t114 - t131 - t135) + t126 + 0.8e1 / 0.11e2 * t113 - 0.7e1 / 0.11e2 * t114) * t112 - 0.24e2 * t70 * t49 * ((t113 * t145) + t117 * (-t114 / 0.2e1 - 0.5e1 / 0.2e1 * t113) + t79 * (-(24 * t113) - 0.17e2 / 0.2e1 * t114) + t33 * (-t155 + 0.15e2 / 0.2e1 * t114 - 0.101e3 / 0.2e1 * t113) + t25 * (t160 + 0.81e2 / 0.2e1 * t114 + (206 * t113)) + (R1 * (t165 - 11 * t114 - t160)) - (32 * t113) - (28 * t114) + t155) * b * t112 + 0.48e2 * t70 * t9 * t220 * t67 - 0.192e3 * t263 * t261 * b * t67 * t1 + 0.240e3 * t70 * t308 * t38 * t66 * t1 - 0.480e3 * t49 * (t176 * (t1 + t314) + t183 * (0.3e1 / 0.5e1 * t1 - t318) + t145 * (-t321 - 0.7e1 / 0.5e1 * t1) + t117 * (-0.13e2 / 0.5e1 * t1 - 0.3e1 / 0.20e2 * t38) + t79 * (t329 - 0.21e2 / 0.5e1 * t1) + t33 * (-(3 * t1) - 0.5e1 / 0.4e1 * t38) + t25 * (-0.11e2 / 0.10e2 * t38 + t338) + R1 * (t318 + t341) - 0.11e2 / 0.20e2 * t38 - 0.24e2 / 0.5e1 * t1) * t55 * t66 * t113 - 0.240e3 * t9 * t386 * t125 * t113 + 0.192e3 * c * t399 * t125 * b * (-0.1e1 / 0.2e1 + 0.11e2 / 0.4e1 * R1 + (2 * t25) - 0.15e2 / 0.4e1 * t33 - (3 * t79) + 0.3e1 / 0.4e1 * t117 + t145) * t177 - (48 * t407 * t405 * t197 * t177 * R1);
t412 = sqrt(t411);
t421 = t70 * t67;
t424 = t1 * t145;
t425 = t1 * t117;
t431 = 0.99e2 / 0.2e1 * t38;
t441 = 8 * t1;
t483 = ((0.3e1 / 0.2e1 * t412 + a * (0.9e1 / 0.2e1 * t263 * t67 * (t25 - R1 + 4) + 0.9e1 / 0.2e1 * t421 * (t25 + 3 * R1 + 10) * b + t49 * (-t424 + (3 * t425) + t79 * (-t329 - (33 * t1)) + t33 * ((97 * t1) + t431) + t25 * (-t431 - (6 * t1)) + R1 * (-(132 * t1) - 0.63e2 / 0.2e1 * t38) + t441 + (36 * t38)) + 0.3e1 * t9 * b * (t424 - t425 + t79 * (-t321 + t441) + t33 * (-t1 + 0.21e2 / 0.2e1 * t38) + t25 * ((5 * t1) - 0.27e2 / 0.2e1 * t38) + R1 * (-(64 * t1) + t321) + (3 * t38) - (44 * t1)) - (3 * c * (22 + 7 * t33 - 13 * t79 + 55 * R1 + 65 * t25 + t117 + t145) * t114) + ((t79 + 2 * t33 - 33 * t25 - 34 * R1 + 1) * (t25 + R1 + 1) * t1 * t55))) * t64 * t63) ^ (0.1e1 / 0.3e1);
t486 = t483 * t62 * t60 / t58 / 0.6e1;
t495 = t25 * t1;
t504 = b * t25;
t509 = t9 * t1;
t512 = t38 * t80 + t9 * t80 - 2 * t11 * t80 + 2 * t38 * t81 - 2 * t9 * t81 + 15 * t38 * t495 - 3 * t38 * t25 * t9 - 6 * b * t51 + 4 * t504 * c * t1 - 3 * t25 * t70 - 15 * t25 * t509;
t513 = R1 * t1;
t518 = c * R1;
t521 = b * t49;
t535 = 14 * t38 * t513 + 6 * t38 * t10 + 18 * t518 * t2 + 12 * R1 * t521 + 6 * R1 * t70 + 28 * R1 * t509 - 3 * t38 * t9 + t114 + 28 * c * t2 - 6 * t521 - 3 * t70 + 4 * t509;
t538 = 2 * c;
t539 = -t518 + t6 + t538;
t540 = t539 ^ 2;
t543 = t62 ^ 2;
t548 = 0.1e1 / t483 * t543 * t58 / t540 * t60 * (t512 + t535) / 0.12e2;
t553 = (1 / t539 * (t504 + t6 - c * t25 + t518 + b + t538)) / 0.3e1;
t555 = sqrt(t23 / 0.4e1 - t32 + t486 + t548 + t553);
t558 = c + b;
t561 = t555 * t7 * a + (t558 * t66 * c) / 0.2e1;
t574 = 0.1e1 / t555;
t577 = sqrt(t23 / 0.2e1 - t32 - t486 - t548 - t553 + t574 * ((t13 * t22 * t30) - (2 * t31 * t13) - (1 / t21 / t20 * t14 * t13) / 0.4e1));
t580 = t558 * t1;
t588 = t66 * t558;
t594 = 4 * t1;
t595 = t38 / 0.4e1;
t610 = -t421 / 0.4e1 - (t49 * t67 * b) / 0.2e1 + t9 * t66 * (t495 + R1 * (-t594 - t595) + t595 + t594) + 0.2e1 * c * b * (t25 - R1 + 0.1e1 / 0.2e1) * t1 * t4 + ((t25 - R1 + 1) * t38 * t513);
t618 = t38 / 0.2e1;
t627 = 2 * t1;
t643 = t398 * t55 * t495;
t650 = -t66 * c + t6;
t676 = t38 / 0.6e1;
R2 = 0.1e1 / (t577 * t650 * t63 * t561 * t1 + t555 * t650 * c * t63 * t66 * t580 / 0.2e1 - t574 * t650 * c * t610 * t588 / 0.2e1 - t7 * ((t263 * t112) / 0.2e1 - (t421 * t4 * b) / 0.2e1 - t49 * t405 * t66 * (t495 + R1 * (-t594 + t618) - t618 + t594) + 0.3e1 * t41 * t66 * (t81 + t25 * (-t676 - 0.4e1 / 0.3e1 * t1) + R1 * (t676 - 0.7e1 / 0.3e1 * t1) + t627) - 0.3e1 * c * (-0.2e1 / 0.3e1 * t25 - 0.7e1 / 0.3e1 * R1 + 0.5e1 / 0.3e1 + t33) * R1 * t114 + t643) * a) * (t577 * t63 * t561 * R1 * t2 + t555 * c * t63 * R1 * b * t66 * t580 / 0.2e1 - t574 * c * t610 * R1 * b * t588 / 0.2e1 - t7 * (-(t421 * t6) / 0.2e1 - 0.2e1 * t49 * (t25 * (t1 + t618) + R1 * (-t594 - t618) + t594) * t66 + t41 * t66 * (t81 + t25 * (t627 - t618) + R1 * (t618 - (12 * t1)) + t441) - 0.2e1 * c * R1 * (-0.9e1 / 0.2e1 * R1 + t33 + 0.3e1) * t114 + t643) * a);
As you can see this is decidedly non-trivial. The mathematical expression goes on for several pages.
The heart of the solutions is that
x = roots( [((2-R1)*c+R1*b)*a, -c*(-1+R1)*(c+b), ((2-R1^2+R1)*c+b*(R1^2+R1+1))*a, -c*(-1+R1)*(c+b), a*(c*R1+b*(R1+1)) ]);
0 件のコメント
Andrei Bobrov
2011 年 8 月 22 日
It solution in MATLAB with Maple Toolbox for MATLAB
>> [R2,x]=solve('((1-R2)/(1-R1)*R1-(c/b*R2))*x^2+(1-R2)/(1-R1)-c/b=0','((2-R1-R2)/(1-R1))*a*x^2+(b+c)*(1-R2)*x+a*(((1-R2)/(1-R1)+R1)+R2)=0','R2,x')
R2 =
[ 2 ]
[ %8 R1 b + b - c + c R1 ]
[ ------------------------------- ]
[ 2 2 2 ]
[ %8 R1 b + %8 c - %8 c R1 + b ]
[ ]
[ 2 ]
[ %9 R1 b + b - c + c R1 ]
[ ------------------------------- ]
[ 2 2 2 ]
[ %9 R1 b + %9 c - %9 c R1 + b ]
[ ]
[ 2 ]
[ %11 R1 b + b - c + c R1 ]
[----------------------------------]
[ 2 2 2 ]
[%11 R1 b + %11 c - %11 c R1 + b]
[ ]
[ 2 ]
[ %12 R1 b + b - c + c R1 ]
[----------------------------------]
[ 2 2 2 ]
[%12 R1 b + %12 c - %12 c R1 + b]
2 3 3 2 2 2 2 2 2 3
%1 := 3 c R1 b - 3 R1 c b - 12 c R1 b + 6 R1 b c + 12 R1 b c + 8 c
3 3 3 2 3 3 3
- R1 c + 6 c R1 - 12 R1 c + R1 b
3 3 2 3 3 2 3 2 2
%2 := (-24 a R1 c b - 168 a R1 c b + 120 a R1 c b
3 2 2 3 2 3 4 2
- 1560 a R1 b c - 1536 a R1 b c + 192 a R1 b c
3 4 2 3 2 4 3 4 2
+ 312 a R1 b c - 1320 a b R1 c + 36 c R1 a b + 180 c R1 b a
3 2 2 4 3 3 2 2 3
- 396 c R1 b a - 612 c R1 b a + 396 c R1 b a - 252 b c a R1
3 2 3 2 2 3 2 3 3 3 3
+ 36 b c a R1 - 324 b c R1 a + 252 b c R1 a + 776 a R1 c
3 2 3 3 3 3 3 3 3 4 3
- 48 a R1 c - 1056 a R1 c - 520 a R1 b - 240 a R1 b
3 4 3 3 3 2 3 2 5 3
- 264 a R1 c - 528 a b R1 - 1056 a b c - 108 c R1 a
5 2 5 2 3 4 3 3
+ 252 c R1 a - 324 c R1 a + 288 b c a + 360 b c a + 64 a c
5 3 3 3 2 3 2
+ 144 c a - 264 a b R1 - 528 a b c + 72 a b c + 12 (
6 5 6 4 2 6 3 3 6 2 4
-96 a b c + 768 a b c - 2304 a b c + 3072 a b c
6 5 4 6 2 4 5 3 4 4 4
- 1536 a b c + 12 a b c - 264 a b c + 492 a b c
4 3 5 4 2 6 4 7 4 8
+ 672 a b c - 1056 a b c - 768 a b c + 192 a c
2 6 4 2 5 5 2 4 6 2 3 7
+ 24 a b c - 96 a b c - 528 a b c - 672 a b c
2 2 8 2 9 2 10 6 6 5 7
- 168 a b c + 192 a b c + 96 a c + 12 b c + 72 b c
4 8 3 9 2 10 11 12
+ 180 b c + 240 b c + 180 b c + 72 b c + 12 c
10 6 6 10 6 5 10 6 4 2
- 48 R1 a b + 192 R1 a b c - 240 R1 a b c
10 6 2 4 10 6 5 10 6 6
+ 240 R1 a b c - 192 R1 a b c + 48 R1 a c
9 6 6 9 6 5 9 6 3 3
- 240 R1 a b + 528 R1 a b c - 480 R1 a b c
9 6 2 4 9 6 5 9 6 6
- 240 R1 a b c + 720 R1 a b c - 288 R1 a c
9 4 6 2 9 4 5 3 9 4 4 4
+ 24 R1 a b c - 24 R1 a b c - 48 R1 a b c
9 4 3 5 9 4 2 6 9 4 7 8 6 6
+ 48 R1 a b c + 24 R1 a b c - 24 R1 a b c - 288 R1 a b
8 6 5 8 6 4 2 8 6 3 3
- 480 R1 a b c + 1632 R1 a b c + 192 R1 a b c
8 6 2 4 8 6 5 8 6 6
- 1248 R1 a b c - 480 R1 a b c + 672 R1 a c
8 4 6 2 8 4 5 3 8 4 4 4
+ 276 R1 a b c + 132 R1 a b c - 816 R1 a b c
8 4 3 5 8 4 2 6 8 4 7
- 264 R1 a b c + 804 R1 a b c + 132 R1 a b c
8 4 8 8 2 6 4 8 2 5 5 8 2 4 6
- 264 R1 a c - 3 R1 a b c - 6 R1 a b c + 3 R1 a b c
8 2 3 7 8 2 2 8 8 2 9 8 2 10
+ 12 R1 a b c + 3 R1 a b c - 6 R1 a b c - 3 R1 a c
7 6 6 7 6 5 7 6 4 2
+ 288 R1 a b - 2400 R1 a b c + 1344 R1 a b c
7 6 3 3 7 6 2 4 7 6 5
+ 960 R1 a b c + 1056 R1 a b c - 96 R1 a b c
7 6 6 7 4 6 2 7 4 5 3
- 1152 R1 a c - 588 R1 a b c + 612 R1 a b c
7 4 4 4 7 4 3 5 7 4 2 6
+ 3024 R1 a b c - 936 R1 a b c - 3996 R1 a b c
7 4 7 7 4 8 7 2 6 4
+ 324 R1 a b c + 1560 R1 a c - 102 R1 a b c
7 2 4 6 7 2 3 7 7 2 2 8
- 258 R1 a b c + 168 R1 a b c + 342 R1 a b c
7 2 9 7 2 10 6 6 6 6 6 5
+ 156 R1 a b c + 18 R1 a c + 624 R1 a b - 576 R1 a b c
6 6 4 2 6 6 3 3 6 6 2 4
- 2928 R1 a b c + 576 R1 a b c + 1488 R1 a b c
6 6 5 6 6 6 6 4 6 2
- 1536 R1 a b c + 2352 R1 a c + 108 R1 a b c
6 4 5 3 6 4 4 4 6 4 3 5
- 648 R1 a b c - 3144 R1 a b c + 1344 R1 a b c
6 4 2 6 6 4 7 6 4 8
+ 6204 R1 a b c - 312 R1 a b c - 2784 R1 a c
6 2 6 4 6 2 5 5 6 2 4 6
+ 447 R1 a b c + 1218 R1 a b c + 621 R1 a b c
6 2 2 8 6 2 9 6 2 10 6 6 6
- 519 R1 a b c + 306 R1 a b c + 219 R1 a c + 12 R1 b c
6 5 7 6 4 8 6 3 9 6 2 10
+ 72 R1 b c + 180 R1 b c + 240 R1 b c + 180 R1 b c
6 11 6 12 5 6 6 5 6 5
+ 72 R1 b c + 12 R1 c - 144 R1 a b + 3312 R1 a b c
5 6 4 2 5 6 3 3 5 6 2 4
- 4320 R1 a b c + 768 R1 a b c - 816 R1 a b c
5 6 5 5 6 6 5 4 6 2
+ 1680 R1 a b c - 3552 R1 a c + 312 R1 a b c
5 4 5 3 5 4 4 4 5 4 3 5
- 2232 R1 a b c + 1320 R1 a b c + 4536 R1 a b c
5 4 2 6 5 4 7 5 4 8
- 5808 R1 a b c - 6912 R1 a b c - 432 R1 a c
5 2 6 4 5 2 5 5 5 2 4 6
- 696 R1 a b c - 1356 R1 a b c + 132 R1 a b c
5 2 3 7 5 2 2 8 5 2 9
+ 168 R1 a b c - 3408 R1 a b c - 4188 R1 a b c
5 2 10 5 6 6 5 5 7 5 4 8
- 1404 R1 a c - 72 R1 b c - 432 R1 b c - 1080 R1 b c
5 3 9 5 2 10 5 11 5 12
- 1440 R1 b c - 1080 R1 b c - 432 R1 b c - 72 R1 c
4 6 6 4 6 5 4 6 4 2
- 480 R1 a b + 1440 R1 a b c + 1536 R1 a b c
4 6 3 3 4 6 2 4 4 6 5
- 576 R1 a b c - 1056 R1 a b c + 288 R1 a b c
4 6 6 4 4 6 2 4 4 5 3
+ 3456 R1 a c - 408 R1 a b c + 2760 R1 a b c
4 4 4 4 4 4 3 5 4 4 2 6
- 1668 R1 a b c - 10332 R1 a b c + 4836 R1 a b c
4 4 7 4 4 8 4 2 6 4
+ 17748 R1 a b c + 7416 R1 a c + 483 R1 a b c
4 2 5 5 4 2 4 6 4 2 3 7
- 162 R1 a b c - 2211 R1 a b c + 2436 R1 a b c
4 2 2 8 4 2 9 4 2 10
+ 11709 R1 a b c + 10974 R1 a b c + 3267 R1 a c
4 6 6 4 5 7 4 4 8 4 3 9
+ 180 R1 b c + 1080 R1 b c + 2700 R1 b c + 3600 R1 b c
4 2 10 4 11 4 12 3 6 6
+ 2700 R1 b c + 1080 R1 b c + 180 R1 c + 96 R1 a b
3 6 5 3 6 4 2 3 6 3 3
- 2208 R1 a b c + 5472 R1 a b c - 4704 R1 a b c
3 6 2 4 3 6 5 3 6 6
+ 960 R1 a b c + 2688 R1 a b c - 3840 R1 a c
3 4 5 3 3 4 4 4 3 4 3 5
- 72 R1 a b c + 300 R1 a b c + 7428 R1 a b c
3 4 2 6 3 4 7 3 4 8
+ 1644 R1 a b c - 14844 R1 a b c - 9864 R1 a c
3 2 6 4 3 2 5 5 3 2 4 6
- 174 R1 a b c + 1524 R1 a b c + 3966 R1 a b c
3 2 3 7 3 2 2 8 3 2 9
- 2856 R1 a b c - 14514 R1 a b c - 13260 R1 a b c
3 2 10 3 6 6 3 5 7 3 4 8
- 3870 R1 a c - 240 R1 b c - 1440 R1 b c - 3600 R1 b c
3 3 9 3 2 10 3 11 3 12
- 4800 R1 b c - 3600 R1 b c - 1440 R1 b c - 240 R1 c
2 6 6 2 6 5 2 6 4 2
+ 144 R1 a b - 576 R1 a b c - 528 R1 a b c
2 6 3 3 2 6 2 4 2 6 5
+ 576 R1 a b c + 2688 R1 a b c - 6912 R1 a b c
2 6 6 2 4 6 2 2 4 5 3
+ 3840 R1 a c - 12 R1 a b c - 636 R1 a b c
2 4 4 4 2 4 3 5 2 4 2 6
+ 3288 R1 a b c - 972 R1 a b c - 8484 R1 a b c
2 4 7 2 4 8 2 2 6 4
+ 2208 R1 a b c + 5808 R1 a c + 105 R1 a b c
2 2 5 5 2 2 4 6 2 2 3 7
- 1338 R1 a b c - 3933 R1 a b c - 252 R1 a b c
2 2 2 8 2 2 9 2 2 10
+ 7887 R1 a b c + 8118 R1 a b c + 2469 R1 a c
2 6 6 2 5 7 2 4 8 2 3 9
+ 180 R1 b c + 1080 R1 b c + 2700 R1 b c + 3600 R1 b c
2 2 10 2 11 2 12 6 6
+ 2700 R1 b c + 1080 R1 b c + 180 R1 c - 48 R1 a b
6 5 6 4 2 6 3 3
+ 720 R1 a b c - 2784 R1 a b c + 4992 R1 a b c
6 2 4 6 5 6 6
- 6144 R1 a b c + 5376 R1 a b c - 1536 R1 a c
4 6 2 4 5 3 4 4 4
- 156 R1 a b c + 372 R1 a b c - 2748 R1 a b c
4 3 5 4 2 6 4 7
- 1524 R1 a b c + 5832 R1 a b c + 2448 R1 a b c
4 8 2 6 4 2 5 5 2 4 6
- 1632 R1 a c - 84 R1 a b c + 540 R1 a b c + 2208 R1 a b c
2 3 7 2 2 8 2 9
+ 1752 R1 a b c - 1332 R1 a b c - 2292 R1 a b c
2 10 6 6 5 7 4 8
- 792 R1 a c - 72 R1 b c - 432 R1 b c - 1080 R1 b c
3 9 2 10 11 12
- 1440 R1 b c - 1080 R1 b c - 432 R1 b c - 72 R1 c
7 2 5 5 6 2 3 7 3 4 6 2 1/2
- 324 R1 a b c - 756 R1 a b c + 432 R1 a b c )
3 2 4 2 3 4 4 4 6 3 2
- 36 a b c R1 - 36 a b c R1 + 36 a c R1 b - 24 R1 a b c
6 3 2 5 3 2 5 3 2 3 3
+ 24 R1 a b c - 24 R1 a b c - 24 R1 a b c + 8 a b
5 4 6 3 3 6 3 3 5 3 3
+ 36 a c R1 + 8 R1 a b - 8 R1 a c + 24 R1 a b
5 3 3 2
+ 24 R1 a c ) %1
%3 := -a R1 c + a R1 b + 2 a c
2 2
%4 := a R1 b + a R1 b - a R1 c + a R1 c + a b + 2 a c
2 2
%5 := -c R1 + b c + c - b c R1
2 1/3
%5 %4 %2 2 4 2 2 4 2 4 2
%6 := 1/4 --- - ---- + 1/6 ----- + 2/3 (a R1 b + a R1 c - 2 R1 a c b
2 %3 %1 a
%3
2 3 2 2 3 2 2 2 2 2 2 2 3 2
+ 2 a R1 b - 2 a R1 c + 15 a R1 b - 3 b c R1 - 6 c R1 b
2 2 4 2 2 2 2 2 2 2 2
+ 4 a R1 c b - 3 c R1 - 15 a R1 c + 14 a R1 b + 6 b c R1
2 3 2 2 4 2 2 2 2
+ 18 a R1 c b + 12 c R1 b + 28 a R1 c + 6 c R1 + a b - 3 b c
2 3 4 2 2 / 2
+ 28 a c b - 6 b c - 3 c + 4 a c ) %1 / ((-c R1 + R1 b + 2 c)
/
2 2
1/3 R1 b + b + R1 b - R1 c + c R1 + 2 c
a %2 ) + 1/3 -------------------------------------
-c R1 + R1 b + 2 c
/
|
|
| 2 1/3
| %5 %4 %2 2 4 2 2 4 2
%7 := |1/2 --- - ---- - 1/6 ----- - 2/3 (a R1 b + a R1 c
| 2 %3 %1 a
\ %3
4 2 2 3 2 2 3 2 2 2 2
- 2 R1 a c b + 2 a R1 b - 2 a R1 c + 15 a R1 b
2 2 2 3 2 2 2 4 2 2 2 2
- 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1 - 15 a R1 c
2 2 2 2 2 3 2 2
+ 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b + 28 a R1 c
4 2 2 2 2 2 3 4 2 2
+ 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c + 4 a c ) %1
/ 2 1/3
/ ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2
%4 %5 %5 %5 |
----- - 2 ---- - 1/4 ---|
2 %3 3|
%3 %3 |
+ ------------------------|
1/2 |
%6 /
%5 1/2
%8 := - 1/4 ---- + 1/2 %6 + 1/2 %7
%3
%5 1/2
%9 := - 1/4 ---- + 1/2 %6 - 1/2 %7
%3
/
|
|
| 2 1/3
| %5 %4 %2 2 4 2 2 4 2
%10 := |1/2 --- - ---- - 1/6 ----- - 2/3 (a R1 b + a R1 c
| 2 %3 %1 a
\ %3
4 2 2 3 2 2 3 2 2 2 2
- 2 R1 a c b + 2 a R1 b - 2 a R1 c + 15 a R1 b
2 2 2 3 2 2 2 4 2 2 2 2
- 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1 - 15 a R1 c
2 2 2 2 2 3 2 2
+ 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b + 28 a R1 c
4 2 2 2 2 2 3 4 2 2
+ 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c + 4 a c ) %1
/ 2 1/3
/ ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2
%4 %5 %5 %5 |
----- - 2 ---- - 1/4 ---|
2 %3 3|
%3 %3 |
- ------------------------|
1/2 |
%6 /
%5 1/2
%11 := - 1/4 ---- - 1/2 %6 + 1/2 %10
%3
%5 1/2
%12 := - 1/4 ---- - 1/2 %6 - 1/2 %10
%3
x =
[ /
[ |
[ |
[ | 2 1/3
[ %5 1/2 | %5 %4 %2
[- 1/4 ---- + 1/2 %6 + 1/2 |1/2 --- - ---- - 1/6 ----- - 2/3 (
[ %3 | 2 %3 %1 a
[ \ %3
2 4 2 2 4 2 4 2 2 3 2 2 3 2
a R1 b + a R1 c - 2 R1 a c b + 2 a R1 b - 2 a R1 c
2 2 2 2 2 2 3 2 2 2 4 2
+ 15 a R1 b - 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1
2 2 2 2 2 2 2 2 3
- 15 a R1 c + 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b
2 2 4 2 2 2 2 2 3 4
+ 28 a R1 c + 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c
2 2 / 2 1/3
+ 4 a c ) %1 / ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2]
%4 %5 %5 %5 | ]
----- - 2 ---- - 1/4 ---| ]
2 %3 3| ]
%3 %3 | ]
+ ------------------------| ]
1/2 | ]
%6 / ]
[ /
[ |
[ |
[ | 2 1/3
[ %5 1/2 | %5 %4 %2
[- 1/4 ---- + 1/2 %6 - 1/2 |1/2 --- - ---- - 1/6 ----- - 2/3 (
[ %3 | 2 %3 %1 a
[ \ %3
2 4 2 2 4 2 4 2 2 3 2 2 3 2
a R1 b + a R1 c - 2 R1 a c b + 2 a R1 b - 2 a R1 c
2 2 2 2 2 2 3 2 2 2 4 2
+ 15 a R1 b - 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1
2 2 2 2 2 2 2 2 3
- 15 a R1 c + 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b
2 2 4 2 2 2 2 2 3 4
+ 28 a R1 c + 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c
2 2 / 2 1/3
+ 4 a c ) %1 / ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2]
%4 %5 %5 %5 | ]
----- - 2 ---- - 1/4 ---| ]
2 %3 3| ]
%3 %3 | ]
+ ------------------------| ]
1/2 | ]
%6 / ]
[ /
[ |
[ |
[ | 2 1/3
[ %5 1/2 | %5 %4 %2
[- 1/4 ---- - 1/2 %6 + 1/2 |1/2 --- - ---- - 1/6 ----- - 2/3 (
[ %3 | 2 %3 %1 a
[ \ %3
2 4 2 2 4 2 4 2 2 3 2 2 3 2
a R1 b + a R1 c - 2 R1 a c b + 2 a R1 b - 2 a R1 c
2 2 2 2 2 2 3 2 2 2 4 2
+ 15 a R1 b - 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1
2 2 2 2 2 2 2 2 3
- 15 a R1 c + 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b
2 2 4 2 2 2 2 2 3 4
+ 28 a R1 c + 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c
2 2 / 2 1/3
+ 4 a c ) %1 / ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2]
%4 %5 %5 %5 | ]
----- - 2 ---- - 1/4 ---| ]
2 %3 3| ]
%3 %3 | ]
- ------------------------| ]
1/2 | ]
%6 / ]
[ /
[ |
[ |
[ | 2 1/3
[ %5 1/2 | %5 %4 %2
[- 1/4 ---- - 1/2 %6 - 1/2 |1/2 --- - ---- - 1/6 ----- - 2/3 (
[ %3 | 2 %3 %1 a
[ \ %3
2 4 2 2 4 2 4 2 2 3 2 2 3 2
a R1 b + a R1 c - 2 R1 a c b + 2 a R1 b - 2 a R1 c
2 2 2 2 2 2 3 2 2 2 4 2
+ 15 a R1 b - 3 b c R1 - 6 c R1 b + 4 a R1 c b - 3 c R1
2 2 2 2 2 2 2 2 3
- 15 a R1 c + 14 a R1 b + 6 b c R1 + 18 a R1 c b + 12 c R1 b
2 2 4 2 2 2 2 2 3 4
+ 28 a R1 c + 6 c R1 + a b - 3 b c + 28 a c b - 6 b c - 3 c
2 2 / 2 1/3
+ 4 a c ) %1 / ((-c R1 + R1 b + 2 c) a %2 )
/
2 2
R1 b + b + R1 b - R1 c + c R1 + 2 c
- 1/3 -------------------------------------
-c R1 + R1 b + 2 c
3\1/2]
%4 %5 %5 %5 | ]
----- - 2 ---- - 1/4 ---| ]
2 %3 3| ]
%3 %3 | ]
- ------------------------| ]
1/2 | ]
%6 / ]
2 3 3 2 2 2 2 2 2 3
%1 := 3 c R1 b - 3 R1 c b - 12 c R1 b + 6 R1 b c + 12 R1 b c + 8 c
3 3 3 2 3 3 3
- R1 c + 6 c R1 - 12 R1 c + R1 b
3 3 2 3 3 2 3 2 2
%2 := (-24 a R1 c b - 168 a R1 c b + 120 a R1 c b
3 2 2 3 2 3 4 2
- 1560 a R1 b c - 1536 a R1 b c + 192 a R1 b c
3 4 2 3 2 4 3 4 2
+ 312 a R1 b c - 1320 a b R1 c + 36 c R1 a b + 180 c R1 b a
3 2 2 4 3 3 2 2 3
- 396 c R1 b a - 612 c R1 b a + 396 c R1 b a - 252 b c a R1
3 2 3 2 2 3 2 3 3 3 3
+ 36 b c a R1 - 324 b c R1 a + 252 b c R1 a + 776 a R1 c
3 2 3 3 3 3 3 3 3 4 3
- 48 a R1 c - 1056 a R1 c - 520 a R1 b - 240 a R1 b
3 4 3 3 3 2 3 2 5 3
- 264 a R1 c - 528 a b R1 - 1056 a b c - 108 c R1 a
5 2 5 2 3 4 3 3
+ 252 c R1 a - 324 c R1 a + 288 b c a + 360 b c a + 64 a c
5 3 3 3 2 3 2
+ 144 c a - 264 a b R1 - 528 a b c + 72 a b c + 12 (
6 5 6 4 2 6 3 3 6 2 4
-96 a b c + 768 a b c - 2304 a b c + 3072 a b c
6 5 4 6 2 4 5 3 4 4 4
- 1536 a b c + 12 a b c - 264 a b c + 492 a b c
4 3 5 4 2 6 4 7 4 8
+ 672 a b c - 1056 a b c - 768 a b c + 192 a c
2 6 4 2 5 5 2 4 6 2 3 7
+ 24 a b c - 96 a b c - 528 a b c - 672 a b c
2 2 8 2 9 2 10 6 6 5 7
- 168 a b c + 192 a b c + 96 a c + 12 b c + 72 b c
4 8 3 9 2 10 11 12
+ 180 b c + 240 b c + 180 b c + 72 b c + 12 c
10 6 6 10 6 5 10 6 4 2
- 48 R1 a b + 192 R1 a b c - 240 R1 a b c
10 6 2 4 10 6 5 10 6 6
+ 240 R1 a b c - 192 R1 a b c + 48 R1 a c
9 6 6 9 6 5 9 6 3 3
- 240 R1 a b + 528 R1 a b c - 480 R1 a b c
9 6 2 4 9 6 5 9 6 6
- 240 R1 a b c + 720 R1 a b c - 288 R1 a c
9 4 6 2 9 4 5 3 9 4 4 4
+ 24 R1 a b c - 24 R1 a b c - 48 R1 a b c
9 4 3 5 9 4 2 6 9 4 7 8 6 6
+ 48 R1 a b c + 24 R1 a b c - 24 R1 a b c - 288 R1 a b
8 6 5 8 6 4 2 8 6 3 3
- 480 R1 a b c + 1632 R1 a b c + 192 R1 a b c
8 6 2 4 8 6 5 8 6 6
- 1248 R1 a b c - 480 R1 a b c + 672 R1 a c
8 4 6 2 8 4 5 3 8 4 4 4
+ 276 R1 a b c + 132 R1 a b c - 816 R1 a b c
8 4 3 5 8 4 2 6 8 4 7
- 264 R1 a b c + 804 R1 a b c + 132 R1 a b c
8 4 8 8 2 6 4 8 2 5 5 8 2 4 6
- 264 R1 a c - 3 R1 a b c - 6 R1 a b c + 3 R1 a b c
8 2 3 7 8 2 2 8 8 2 9 8 2 10
+ 12 R1 a b c + 3 R1 a b c - 6 R1 a b c - 3 R1 a c
7 6 6 7 6 5 7 6 4 2
+ 288 R1 a b - 2400 R1 a b c + 1344 R1 a b c
7 6 3 3 7 6 2 4 7 6 5
+ 960 R1 a b c + 1056 R1 a b c - 96 R1 a b c
7 6 6 7 4 6 2 7 4 5 3
- 1152 R1 a c - 588 R1 a b c + 612 R1 a b c
7 4 4 4 7 4 3 5 7 4 2 6
+ 3024 R1 a b c - 936 R1 a b c - 3996 R1 a b c
7 4 7 7 4 8 7 2 6 4
+ 324 R1 a b c + 1560 R1 a c - 102 R1 a b c
7 2 4 6 7 2 3 7 7 2 2 8
- 258 R1 a b c + 168 R1 a b c + 342 R1 a b c
7 2 9 7 2 10 6 6 6 6 6 5
+ 156 R1 a b c + 18 R1 a c + 624 R1 a b - 576 R1 a b c
6 6 4 2 6 6 3 3 6 6 2 4
- 2928 R1 a b c + 576 R1 a b c + 1488 R1 a b c
6 6 5 6 6 6 6 4 6 2
- 1536 R1 a b c + 2352 R1 a c + 108 R1 a b c
6 4 5 3 6 4 4 4 6 4 3 5
- 648 R1 a b c - 3144 R1 a b c + 1344 R1 a b c
6 4 2 6 6 4 7 6 4 8
+ 6204 R1 a b c - 312 R1 a b c - 2784 R1 a c
6 2 6 4 6 2 5 5 6 2 4 6
+ 447 R1 a b c + 1218 R1 a b c + 621 R1 a b c
6 2 2 8 6 2 9 6 2 10 6 6 6
- 519 R1 a b c + 306 R1 a b c + 219 R1 a c + 12 R1 b c
6 5 7 6 4 8 6 3 9 6 2 10
+ 72 R1 b c + 180 R1 b c + 240 R1 b c + 180 R1 b c
6 11 6 12 5 6 6 5 6 5
+ 72 R1 b c + 12 R1 c - 144 R1 a b + 3312 R1 a b c
5 6 4 2 5 6 3 3 5 6 2 4
- 4320 R1 a b c + 768 R1 a b c - 816 R1 a b c
5 6 5 5 6 6 5 4 6 2
+ 1680 R1 a b c - 3552 R1 a c + 312 R1 a b c
5 4 5 3 5 4 4 4 5 4 3 5
- 2232 R1 a b c + 1320 R1 a b c + 4536 R1 a b c
5 4 2 6 5 4 7 5 4 8
- 5808 R1 a b c - 6912 R1 a b c - 432 R1 a c
5 2 6 4 5 2 5 5 5 2 4 6
- 696 R1 a b c - 1356 R1 a b c + 132 R1 a b c
5 2 3 7 5 2 2 8 5 2 9
+ 168 R1 a b c - 3408 R1 a b c - 4188 R1 a b c
5 2 10 5 6 6 5 5 7 5 4 8
- 1404 R1 a c - 72 R1 b c - 432 R1 b c - 1080 R1 b c
5 3 9 5 2 10 5 11 5 12
- 1440 R1 b c - 1080 R1 b c - 432 R1 b c - 72 R1 c
4 6 6 4 6 5 4 6 4 2
- 480 R1 a b + 1440 R1 a b c + 1536 R1 a b c
4 6 3 3 4 6 2 4 4 6 5
- 576 R1 a b c - 1056 R1 a b c + 288 R1 a b c
4 6 6 4 4 6 2 4 4 5 3
+ 3456 R1 a c - 408 R1 a b c + 2760 R1 a b c
4 4 4 4 4 4 3 5 4 4 2 6
- 1668 R1 a b c - 10332 R1 a b c + 4836 R1 a b c
4 4 7 4 4 8 4 2 6 4
+ 17748 R1 a b c + 7416 R1 a c + 483 R1 a b c
4 2 5 5 4 2 4 6 4 2 3 7
- 162 R1 a b c - 2211 R1 a b c + 2436 R1 a b c
4 2 2 8 4 2 9 4 2 10
+ 11709 R1 a b c + 10974 R1 a b c + 3267 R1 a c
4 6 6 4 5 7 4 4 8 4 3 9
+ 180 R1 b c + 1080 R1 b c + 2700 R1 b c + 3600 R1 b c
4 2 10 4 11 4 12 3 6 6
+ 2700 R1 b c + 1080 R1 b c + 180 R1 c + 96 R1 a b
3 6 5 3 6 4 2 3 6 3 3
- 2208 R1 a b c + 5472 R1 a b c - 4704 R1 a b c
3 6 2 4 3 6 5 3 6 6
+ 960 R1 a b c + 2688 R1 a b c - 3840 R1 a c
3 4 5 3 3 4 4 4 3 4 3 5
- 72 R1 a b c + 300 R1 a b c + 7428 R1 a b c
3 4 2 6 3 4 7 3 4 8
+ 1644 R1 a b c - 14844 R1 a b c - 9864 R1 a c
3 2 6 4 3 2 5 5 3 2 4 6
- 174 R1 a b c + 1524 R1 a b c + 3966 R1 a b c
3 2 3 7 3 2 2 8 3 2 9
- 2856 R1 a b c - 14514 R1 a b c - 13260 R1 a b c
3 2 10 3 6 6 3 5 7 3 4 8
- 3870 R1 a c - 240 R1 b c - 1440 R1 b c - 3600 R1 b c
3 3 9 3 2 10 3 11 3 12
- 4800 R1 b c - 3600 R1 b c - 1440 R1 b c - 240 R1 c
2 6 6 2 6 5 2 6 4 2
+ 144 R1 a b - 576 R1 a b c - 528 R1 a b c
2 6 3 3 2 6 2 4 2 6 5
+ 576 R1 a b c + 2688 R1 a b c - 6912 R1 a b c
2 6 6 2 4 6 2 2 4 5 3
+ 3840 R1 a c - 12 R1 a b c - 636 R1 a b c
2 4 4 4 2 4 3 5 2 4 2 6
+ 3288 R1 a b c - 972 R1 a b c - 8484 R1 a b c
2 4 7 2 4 8 2 2 6 4
+ 2208 R1 a b c + 5808 R1 a c + 105 R1 a b c
2 2 5 5 2 2 4 6 2 2 3 7
- 1338 R1 a b c - 3933 R1 a b c - 252 R1 a b c
2 2 2 8 2 2 9 2 2 10
+ 7887 R1 a b c + 8118 R1 a b c + 2469 R1 a c
2 6 6 2 5 7 2 4 8 2 3 9
+ 180 R1 b c + 1080 R1 b c + 2700 R1 b c + 3600 R1 b c
2 2 10 2 11 2 12 6 6
+ 2700 R1 b c + 1080 R1 b c + 180 R1 c - 48 R1 a b
6 5 6 4 2 6 3 3
+ 720 R1 a b c - 2784 R1 a b c + 4992 R1 a b c
6 2 4 6 5 6 6
- 6144 R1 a b c + 5376 R1 a b c - 1536 R1 a c
4 6 2 4 5 3 4 4 4
- 156 R1 a b c + 372 R1 a b c - 2748 R1 a b c
4 3 5 4 2 6 4 7
- 1524 R1 a b c + 5832 R1 a b c + 2448 R1 a b c
4 8 2 6 4 2 5 5 2 4 6
- 1632 R1 a c - 84 R1 a b c + 540 R1 a b c + 2208 R1 a b c
2 3 7 2 2 8 2 9
+ 1752 R1 a b c - 1332 R1 a b c - 2292 R1 a b c
2 10 6 6 5 7 4 8
- 792 R1 a c - 72 R1 b c - 432 R1 b c - 1080 R1 b c
3 9 2 10 11 12
- 1440 R1 b c - 1080 R1 b c - 432 R1 b c - 72 R1 c
7 2 5 5 6 2 3 7 3 4 6 2 1/2
- 324 R1 a b c - 756 R1 a b c + 432 R1 a b c )
3 2 4 2 3 4 4 4 6 3 2
- 36 a b c R1 - 36 a b c R1 + 36 a c R1 b - 24 R1 a b c
6 3 2 5 3 2 5 3 2 3 3
+ 24 R1 a b c - 24 R1 a b c - 24 R1 a b c + 8 a b
5 4 6 3 3 6 3 3 5 3 3
+ 36 a c R1 + 8 R1 a b - 8 R1 a c + 24 R1 a b
5 3 3 2
+ 24 R1 a c ) %1
%3 := -a R1 c + a R1 b + 2 a c
2 2
%4 := a R1 b + a R1 b - a R1 c + a R1 c + a b + 2 a c
2 2
%5 := -c R1 + b c + c - b c R1
2 1/3
%5 %4 %2 2 4 2 2 4 2 4 2
%6 := 1/4 --- - ---- + 1/6 ----- + 2/3 (a R1 b + a R1 c - 2 R1 a c b
2 %3 %1 a
%3
2 3 2 2 3 2 2 2 2 2 2 2 3 2
+ 2 a R1 b - 2 a R1 c + 15 a R1 b - 3 b c R1 - 6 c R1 b
2 2 4 2 2 2 2 2 2 2 2
+ 4 a R1 c b - 3 c R1 - 15 a R1 c + 14 a R1 b + 6 b c R1
2 3 2 2 4 2 2 2 2
+ 18 a R1 c b + 12 c R1 b + 28 a R1 c + 6 c R1 + a b - 3 b c
2 3 4 2 2 / 2
+ 28 a c b - 6 b c - 3 c + 4 a c ) %1 / ((-c R1 + R1 b + 2 c)
/
2 2
1/3 R1 b + b + R1 b - R1 c + c R1 + 2 c
a %2 ) + 1/3 -------------------------------------
-c R1 + R1 b + 2 c
>>
4 件のコメント
Walter Roberson
2011 年 8 月 23 日
It used to be that the Symbolic Math toolbox was based on Maple; now it is based upon MuPad.
After Mathworks bought MuPad, there were a few generations where if you owned Maple you could use it as the engine for the symbolic toolbox. Recent release notes have left me confused about whether that is still supported.
Maple is from Maplesoft, and it is *not* free: it is priced at around $2000 or so.
The formatted symbolic solution that Andrei posted can be made much shorter. I will need to use an Answer to do the proper formatting though.
Walter Roberson
2011 年 8 月 23 日
It used to be that the Symbolic Math toolbox was based on Maple; now it is based upon MuPad.
After Mathworks bought MuPad, there were a few generations where if you owned Maple you could use it as the engine for the symbolic toolbox. Recent release notes have left me confused about whether that is still supported.
Maple is from Maplesoft, and it is *not* free: it is priced at around $2000 or so.
The formatted symbolic solution that Andrei posted can be made much shorter. I will need to use an Answer to do the proper formatting though.
Walter Roberson
2011 年 8 月 23 日
The more compact formatted Maple output:
2
%2 R1 b + b - c + c R1
[[R2 = -------------------------------, x = %2]]
2 2 2
%2 R1 b + %2 c - %2 c R1 + b
2 2
%1 := -c R1 + b c + c - b c R1
4 3
%2 := RootOf((-a R1 c + a R1 b + 2 a c) _Z + %1 _Z
2 2 2
+ (a R1 b + a R1 b - a R1 c + a R1 c + a b + 2 a c) _Z + %1 _Z
+ a R1 c + a b + a R1 b)
The %1 and %2 are like temporary variables just to make the printout shorter.
In Maple, RootOf(some expression in _Z) stands in for the list of _Z such that the expression evaluates to 0 -- in other words, stands in for the roots of the expression. MuPad has almost exactly the same thing, except that instead of using _Z as the dummy variable, MuPad generates the dummy variable name dynamically (e.g., it might be X17 on one run) and MuPad puts the name of the dummy variable in as the second parameter of RootOf(). Thus in MuPad, you examine the second parameter of the RootOf first, as that will tell you the name of the dummy variable, and then the first parameter tells you the expression that you will substitute particular values of the dummy variable in to in order to make the expression 0.
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