第9章 符号运算
第9章 符号运算
9.1 符号运算基础
9.1.1 创建符号对象和表达式
9.1.1.1 sym函数
S=sym(A) % 创建符号矩阵/变量
S=sym('A',[n1...nM]) % 创建n1*...*nM的符号矩阵
S=sym('A',n) % 创建n行n列的符号矩阵
S=sym(___,set) % set设置创建的符号变量或数组元素的属性
S=sym(___,'clear') % clear清除所有以前对变量x的设置
S=sym(A,flag) % flag为转换的符号对象应该符合的格式
其中，set可以为'real'（实数）、'positive'（正数）、'integer'（整数）、'rational'（有理数）。当被转换的对象为数值对象时，flag可以有如下选择：
- r：为默认设置，最接近有理表示的形式；
- d：最接近的十进制浮点精确表示；
- e：带数值计算时估计误差的有理表示；
- f：十六进制浮点表示。
>> var=sym('x')
var =
x
>> num=sym(8)
num =
8
>> a=sym('x_%d',[1 4])
a =
[x_1, x_2, x_3, x_4]
>> A=sym('a',[2 2 2])
A(:,:,1) =
[a1_1_1, a1_2_1]
[a2_1_1, a2_2_1]
A(:,:,2) =
[a1_1_2, a1_2_2]
[a2_1_2, a2_2_2]
9.1.1.2 syms函数
syms var1 ... varN % 创建多个符号对象
syms f(varl,...,varN) % 创建符号函数f或符号变量var1,...,varN
syms ___ [n1 ... nM] % 创建n1*...*nM的符号对象或符号函数
syms ___ n % 创建n*n的符号对象或符号函数
syms ___ set % set设置创建符号函数f或符号变量的属性
>> syms x y z % 创建符号对象变量x、y、z
>> syms a [1 4] % 创建符号向量
>> a
a =
[a1, a2, a3, a4]
>> syms 'p_a%d' 'p_b%d' [1 4] % 创建符号向量
>> p_a
p_a =
[p_a1, p_a2, p_a3, p_a4]
>> p_b
p_b =
[p_b1, p_b2, p_b3, p_b4]
>> syms A [3 4] % 创建符号矩阵
>> A
A =
[A1_1, A1_2, A1_3, A1_4]
[A2_1, A2_2, A2_3, A2_4]
[A3_1, A3_2, A3_3, A3_4]
>> syms s(t) f(x,y) % 创建符号函数s(t)、f(x,y)
>> f(x,y)=x+2*y % 定义函数
f(x, y) =
x + 2*y
>> f(1,2)
ans =
5
>> syms x
>> M=[x x^3; x^2 x^4]; % 用矩阵作为公式创建和计算符号函数
>> f(x)=M
f(x) =
[ x, x^3]
[x^2, x^4]
>> f(4)
ans =
[ 4, 64]
[16, 256]
9.1.1.3 class函数
className=class(obj) % 返回obj对象数据的类型
>> a=2; b='2';
>> c=sym(2); d=sym('2');
>> classa=class(a)
classa =
'double' % double为双精度类型
>> classb=class(b)
classb =
'char' % char为字符型
>> classc=class(c)
classc =
'sym' % sym为符号型
>> classd=class(d)
classd =
'sym'
>> a=sym('a');
>> classa=class(a)
classa =
'sym'
>> syms a;
>> classa=class(a)
classa =
'sym'
9.1.2 符号对象的基本运算
>> syms x y z;
>> f1=x^2+y^2+z^2+1
f1 =
x^2 + y^2 + z^2 + 1
>> classf1=class(f1)
classf1 =
'sym'
>> syms a b c d e f;
>> m1=[a b c; d e f]
m1 =
[a, b, c]
[d, e, f]
>> classm1=class(m1)
classm1 =
'sym'
9.1.3 符号表达式的替换
9.1.3.1 subexpr函数
[Y,sigma]=subexpr(S,sigma) % 指定用变量sigma的值（必须为符号对象）替换；符号表达式（可以是矩阵）中重复出现的字符串
[Y,sigma]=subexpr(S,sigma) % 输入参数sigma是字符或字符串
>> syms a b c x;
>> s=solve(a*x^2+b*x+c==0)
s =
-(b + (b^2 - 4*a*c)^(1/2))/(2*a)
-(b - (b^2 - 4*a*c)^(1/2))/(2*a)
>> r=subexpr(s) % 用字符串代替相同部分
sigma =
(b^2 - 4*a*c)^(1/2)
r =
-(b + sigma)/(2*a)
-(b - sigma)/(2*a)
9.1.3.2 subs函数
R=subs(S) % 用工作空间中的变量值替代符号表达式S中的所有符号变量；如果没有指定某符号变量的值，则返回值中该符号变量不被替换
R=subs(S,New) % 用新符号变量New替代原来符号表达式S中的默认变量；确定默认变量的规则与findsym函数的规则相同
R=subs(S,Old,New) % 用新符号变量New替代原来符号表达式S中的变量O1d
>> syms a b t;
>> subs(a^2+a*b+8,a,1) % 简单替换，将a+b中的a替换为1
ans =
b + 9
>> subs(exp(a*t),'a',-magic(2)) % 用矩阵替换符号变量
ans =
[ exp(-t), exp(-3*t)]
[exp(-4*t), exp(-2*t)]
9.1.3.3 pretty函数
>> syms a x
>> s=solve(x^2+x+a)
s =
- (1 - 4*a)^(1/2)/2 - 1/2
(1 - 4*a)^(1/2)/2 - 1/2
>> pretty(s)
/ sqrt(1 - 4 a) 1 \
| - ------------- - - |
| 2 2 |
| |
| sqrt(1 - 4 a) 1 |
| ------------- - - |
\ 2 2 /
9.1.4 符号表达式的简化
9.1.4.1 collect函数
R=collect(S) % 合并表达式S中相同次幂的项。S可以是表达式，也可以是符号矩阵
R=collect(S,v) % 合并表达式S中具有v次幂的项。不指定v，则合并所有x相同次幂的项
>> syms x y
>> coeffs=collect((exp(x)+x)*(x+2))
coeffs =
x^2 + (exp(x) + 2)*x + 2*exp(x)
>> coeffs_x=collect(x^2*y+y*x-x^2-2*x,x)
coeffs_x =
(y - 1)*x^2 + (y - 2)*x
>> coeffs_y=collect(x^2*y+y*x-x^2-2*x,y)
coeffs_y =
(x^2 + x)*y - x^2 - 2*x
>> syms a b
>> coeffs_xy=collect(a^2*x*y+a*b*x^2+a*x*y+x^2,[x y])
coeffs_xy =
(a*b + 1)*x^2 + (a^2 + a)*x*y
9.1.4.2 expand函数
R=expand(S)
>> syms x y
>> expand(cos(x+y)) % 将三角函数展开
ans =
cos(x)*cos(y) - sin(x)*sin(y)
>> expand((x^2+x+y+1)^2) % 将多项式展开
ans =
x^4 + 2*x^3 + 2*x^2*y + 3*x^2 + 2*x*y + 2*x + y^2 + 2*y + 1
>> expand(exp(x+y+2)) % 指数函数的展开
ans =
exp(2)*exp(x)*exp(y)
9.1.4.3 horner函数
R=horner(S) % S是符号多项式矩阵，将其中每个多项式都转换成它们的嵌套形式
>> syms x y;
>> f=x^3-6*x^2+11*x-6;
>> horner(f)
ans =
x*(x*(x - 6) + 11) - 6
9.1.4.4 factor函数
f=factor(n) % 返回包含n的质因数的行向量，向量f与n具有相同的数据类型
f=factor(X) % 把X表示成系数为有理数的低阶多项式相乘的形式，X为多项式，系数为有理数；若X不能分解成有理多项式乘积的形式，则返回X本身
>> f=factor(98) % 求98的质因数
f =
2 7 7
>> syms x y n;
>> f=2*x^2-7*x*y-5*x-22*y^2+35*y-3;
>> factor(f) % 对多项式进行因式分解
ans =
[2*x - 11*y + 1, x + 2*y - 3]
9.1.4.5 simplify函数
R=simplify(A)
>> syms x;
>> S=sym((x^2-x-2)/(x+1));
>> simplify(S)
ans =
x - 2
>> M=[(x^2+5*x+6)/(x+2), sin(x)*sin(2*x)+cos(x)*cos(2*x); (exp(-x*1i)*1i)/2-(exp(x*1i)*1i)/2, sqrt(16)];
>> S=simplify(M)
S =
[ x + 3, cos(x)]
[sin(x), 4]
9.1.5 精度计算
digits(d) % 将近似解的精度调整为d位有效数字，默认为32，为空时得到当前采用的精度
vpa(A,d) % 求符号解A的近似解，该近似解的有效位数由参数d指定；如果不指定d，则按照一个digits(d)指令设置的有效位数输出
double(A) % 把符号矩阵或任意精度表示的矩阵A转换成双精度矩阵
>> A=[3.100 1.300 5.500; 4.970 4.400 1; 9.000 2.90 4.61];
>> S=sym(A)
S =
[ 31/10, 13/10, 11/2]
[497/100, 22/5, 1]
[ 9, 29/10, 461/100]
>> digits(6) % 转换成有效位数为6的任意精度的矩阵
>> vpa(S)
ans =
[ 3.1, 1.3, 5.5]
[4.97, 4.4, 1.0]
[ 9.0, 2.9, 4.61]
>> double(S) % 转换成双精度矩阵
ans =
3.1000 1.3000 5.5000
4.9700 4.4000 1.0000
9.0000 2.9000 4.6100
9.2 符号微积分及其变换
9.2.1 符号表达式的微分运算
9.2.1.1 diff函数
Y=diff(X) % 对符号表达式或符号矩阵X求微分
Y=diff(X,n) % 对X中的默认变量进行n阶微分运算
Y=diff(X,n,dim) % 对符号表达式或矩阵X沿dim指定的维进行n阶微分运算
>> syms a x
>> f=tan(x);
>> df=diff(f)
df =
tan(x)^2 + 1
>> df=diff(f,2)
df =
2*tan(x)*(tan(x)^2 + 1)
>> syms a t x;
>> f=[a, t; t*sin(x), log(x)];
>> df=diff(f)
df =
[ 0, 0]
[t*cos(x), 1/x]
>> dfdxdt=diff(diff(f,x),t)
dfdxdt =
[ 0, 0]
[cos(x), 0]
9.2.1.2 Jacobian函数
设 \(\displaystyle F(x_1,x_2,\dots,x_n) = \begin{pmatrix} f_1(x_1,x_2,\dots,x_n) \\ f_2(x_1,x_2,\dots,x_n) \\ \vdots \\ f_n(x_1,x_2,\dots,x_n) \\ \end{pmatrix}\)，其Jacobian矩阵的数学表达式为\(\displaystyle J = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \dfrac{\partial f_2}{\partial x_1} & \cdots & \dfrac{\partial f_2}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_n}{\partial x_1} & \cdots & \dfrac{\partial f_n}{\partial x_n} \end{pmatrix}\)。
R=jacobian(f,v) % f是一个符号列向量，v是指定进行变换的变量组成的行向量
>> syms x1 x2;
>> f=[exp(x1); sin(x2); cos(x1)];
>> v=[x1 x2];
>> fjac=jacobian(f,v)
fjac =
[ exp(x1), 0]
[ 0, cos(x2)]
[-sin(x1), 0]
9.2.1.3 符号表达式的极限
limit(F,x,a) % 求当x→a时符号表达式F的极限
limit(F,a) % F采用默认自变量，求F的自变量趋近于a时的极限值
limit(F) % F采用默自变量，并以a=0作为自变量的趋近值，求F的极限值
limit(F,x,a,'left') % 求F的左极限，即自变量从左边趋近于a时的函数极限值
limit(F,x,a,'right') % 求F的右极限，即自变量从右边趋近于a时的函数极限值
>> syms x;
>> limit((x+1)/x^3,x,0)
ans =
NaN
>> limit((x+1)/x^3,x,0,'left')
ans =
-Inf
>> limit((x+1)/x^3,x,0,'right')
ans =
Inf
9.2.2 符号表达式的级数与积分
9.2.2.1 级数求和
r=symsum(s,v,a,b) % 求符号表达式s中的变量v从a到b的和
r=symsum(s,a,b) % 求符号表达式s中的默认自变量从a到b的和
r=symsum(s,v) % 求符号表达式s中的变量v从0到v-1的和
>> syms k x;
>> r=symsum(k^2,1,10)
r =
385
>> F1=symsum(k,k)
F1 =
k^2/2 - k/2
>> F2=symsum(2^k,k)
F2 =
2^k
>> F(x)=symsum(k*x^k,k,1,8)
F(x) =
8*x^8 + 7*x^7 + 6*x^6 + 5*x^5 + 4*x^4 + 3*x^3 + 2*x^2 + x
>> F(2)
ans =
3586
9.2.2.2 Taylor级数
T=taylor(f) % 返回符号表达式f在默认变量等于0处做5阶Taylor展开时的展开式
T=taylor(f,v) % 返回符号表达式f在v=0处做5阶Taylor展开时的展开式
T=taylor(f,v,a) % 返回f在v=a处做5阶Taylor展开的展开式
T=taylor(f,v,'Order',n) % 返回f的n-1阶麦克劳林级数展开式即在v=0处做Taylor展开，f以符号标量v作为自变量
>> syms x;
>> t=taylor(exp(x))
t =
x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1
>> t=taylor(sin(x))
t =
x^5/120 - x^3/6 + x
>> T=taylor(1/exp(x)-exp(x)+2*x,x,'Order',5)
T =
-x^3/3
9.2.2.3 符号积分
R=int(S) % 用默认变量（可用函数findsym确定）求符号表达式S的不定积分值
R=int(S,v) % 用符号标量v作为变量求符号表达式S的不定积分值
R=int(S,a,b) % 用来求默认变量从a变到b时的符号表达式
R=int(S,v,a,b) % 求当v从a变到b时符号表达式S的定积分值，S采用符号标量v作为变量
>> syms x;
>> int(sin(x))
ans =
-cos(x)
>> int(sin(x),0,pi)
ans =
2
>> syms x y;
>> int(int(x^2+y^2,x,x^2),x,y)
ans =
- x^7/21 + x^4/12 - (x^3*y^2)/3 + (x^2*y^2)/2 + y^7/21 + y^5/3 - (7*y^4)/12
9.2.3 符号积分变换
9.2.3.1 Fourier变换及其逆变换
\[F(\omega) = \int_{- \infty}^{\infty} f(t) \mathrm{e}^{\mathrm{i} \omega t} \,\mathrm{d} t \longleftrightarrow f(t) = \int_{- \infty}^{\infty} F(\omega) \mathrm{e}^{\mathrm{i} \omega t} \,\mathrm{d} \omega
\]
Fw=fourier(ft,t,w) % 求时域函数ft的Fourier变换Fw
ft=ifourier(Fw,w,t) % 求频域函数Fw的Eourier逆变换ft
>> syms t w
>> ut=sym(heaviside(t)); % heaviside为单位阶跃函数
>> UT=fourier(ut,t,w)
UT =
pi*dirac(w) - 1i/w
>> Ut=ifourier(UT,w,t)
Ut =
(pi + pi*sign(t))/(2*pi)
9.2.3.2 Laplace变换及其逆变换
\[F(s) = \int_{0}^{\infty} f(t) \mathrm{e}^{- s t} \,\mathrm{d} t \longleftrightarrow f(t) = \dfrac{1}{2 \pi \mathrm{i}} \int_{c - \mathrm{i} \infty}^{c + \mathrm{i} \infty} F(s) \mathrm{e}^{s t} \,\mathrm{d} s
\]
Fs=laplace(ft,t,s) % 求时域函数ft的Laplace变换Fs
ft=ilaplace(Fs,s,t) % 求频域函数Fs的Laplace逆变换ft
>> syms t s
>> syms a b positive
>> Mt=[dirac(t-a), heaviside(t-b); exp(-t)*sin(b*t), cos(t)]; % dirac和heaviside分别为单位脉冲函数和单位阶跃函数
>> MS=laplace(Mt,t,s)
MS =
[ exp(-a*s), exp(-b*s)/s]
[b/((s + 1)^2 + b^2), s/(s^2 + 1)]
>> ft=ilaplace(MS,s,t)
ft =
[ dirac(a - t), heaviside(t - b)]
[exp(-t)*sin(b*t), cos(t)]
9.2.3.3 Z变换及其逆变换
\[F(\mathrm{z}) = \sum_{n=0}^{\infty} f(n) \mathrm{z}^{-n} \longleftrightarrow f(n) = Z^{-1}\{ F(\mathrm{z}) \}
\]
FZ=ztrans(fn,n,z) % 求时域函数fn的Z变换FZ
fn=iztrans(Fz,z,n) % 求频域函数FZ的Z逆变换fn
>> syms a b t z n
>> f=1/(a-b)*(exp(-(b*t))-exp(-a*t));
>> Fz=ztrans(f)
Fz =
z/((z - exp(-b))*(a - b)) - z/((z - exp(-a))*(a - b))
>> fn=iztrans(Fz,z,n)
fn =
- (exp(-a)*(exp(-a)^n*exp(a) - exp(a)*kroneckerDelta(n, 0)))/(a - b) - (exp(-b)*(exp(b)*kroneckerDelta(n, 0) - exp(-b)^n*exp(b)))/(a - b)
9.3 符号矩阵的计算
9.3.1 代数基本运算
>> syms a b c d
>> A=sym([a b; c d]); % 定义符号矩阵
>> B=sym([2*a b; c 2*d]); % 定义符号矩阵
>> A+B
ans =
[3*a, 2*b]
[2*c, 3*d]
>> A*B
ans =
[2*a^2 + b*c, a*b + 2*b*d]
[2*a*c + c*d, 2*d^2 + b*c]
9.3.2 线性代数运算
>> H=hilb(6); % 生成六阶希尔伯特数值矩阵
>> H=sym(H) % 将数值矩阵转换成符号矩阵
H =
[ 1, 1/2, 1/3, 1/4, 1/5, 1/6]
[1/2, 1/3, 1/4, 1/5, 1/6, 1/7]
[1/3, 1/4, 1/5, 1/6, 1/7, 1/8]
[1/4, 1/5, 1/6, 1/7, 1/8, 1/9]
[1/5, 1/6, 1/7, 1/8, 1/9, 1/10]
[1/6, 1/7, 1/8, 1/9, 1/10, 1/11]
>> inv(H) % 求符号矩阵的逆矩阵
ans =
[ 36, -630, 3360, -7560, 7560, -2772]
[ -630, 14700, -88200, 211680, -220500, 83160]
[ 3360, -88200, 564480, -1411200, 1512000, -582120]
[-7560, 211680, -1411200, 3628800, -3969000, 1552320]
[ 7560, -220500, 1512000, -3969000, 4410000, -1746360]
[-2772, 83160, -582120, 1552320, -1746360, 698544]
>> det(H) 方阵H的行列式的值
ans =
1/186313420339200000
9.3.3 特征值分解
E=eig(A) % 求符号方阵A的符号特征值E
[v,E]=eig(A) % 返回方阵A的符号特征值E和相应的特征向量v
>> H=hilb(6); % 生成六阶希尔伯特数值矩阵
>> H=sym(H); % 将数值矩阵转换成符号矩阵
>> [v,E]=eig(H) % v的每一列是H的一个特征向量，E的对角线元素是H的特征值
v =
[ (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1))/9904125 - 2162603/445685625, (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2))/9904125 - 2162603/445685625, (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3))/9904125 - 2162603/445685625, (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4))/9904125 - 2162603/445685625, (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5))/9904125 - 2162603/445685625, (3705925043136*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^3)/13475 - (34849407392*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^2)/8085 - (13407123456*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^4)/11 + 648953856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^5 + (23483572087*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6))/9904125 - 2162603/445685625]
[ (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1))/29712375 + 362980853/2674113750, (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2))/29712375 + 362980853/2674113750, (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3))/29712375 + 362980853/2674113750, (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4))/29712375 + 362980853/2674113750, (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5))/29712375 + 362980853/2674113750, (1809095551216*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^2)/24255 - (64002961369856*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^3)/13475 + (231516896256*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^4)/11 - 11206066176*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^5 - (1273968333956*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6))/29712375 + 362980853/2674113750]
[(9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1))/346644375 - 1284341752/1418090625, (9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2))/346644375 - 1284341752/1418090625, (9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3))/346644375 - 1284341752/1418090625, (9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4))/346644375 - 1284341752/1418090625, (9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5))/346644375 - 1284341752/1418090625, (9162342482196864*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^3)/471625 - (7856710738528*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^2)/25725 - (33140332652544*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^4)/385 + (1604068614144*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^5)/35 + (62594766554788*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6))/346644375 - 1284341752/1418090625]
[ (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1))/23109625 + 808317061/346644375, (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2))/23109625 + 808317061/346644375, (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3))/23109625 + 808317061/346644375, (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4))/23109625 + 808317061/346644375, (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5))/23109625 + 808317061/346644375, (8381592314652*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^2)/18865 - (2663780355756288*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^3)/94325 + (9634462195968*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^4)/77 - (466327093248*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^5)/7 - (6191864276632*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6))/23109625 + 808317061/346644375]
[ (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1))/207986625 - 724286342/283618125, (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2))/207986625 - 724286342/283618125, (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3))/207986625 - 724286342/283618125, (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4))/207986625 - 724286342/283618125, (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5))/207986625 - 724286342/283618125, (1265964151819744*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^3)/94325 - (11956428602368*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^2)/56595 - (4578627354624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^4)/77 + (221613594624*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)^5)/7 + (26892406032994*root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6))/207986625 - 724286342/283618125]
[ 1, 1, 1, 1, 1, 1]
E =
[root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 1), 0, 0, 0, 0, 0]
[ 0, root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 2), 0, 0, 0, 0]
[ 0, 0, root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 3), 0, 0, 0]
[ 0, 0, 0, root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 4), 0, 0]
[ 0, 0, 0, 0, root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 5), 0]
[ 0, 0, 0, 0, 0, root(z^6 - (6508*z^5)/3465 + (14806217*z^4)/34927200 - (18344719*z^3)/2750517000 + (10828423*z^2)/2688505344000 - (3529*z)/70573265280000 + 1/186313420339200000, z, 6)]
9.3.4 约当标准型
J=jordan(A) % 计算矩阵A的约当标准型。其中A可以是数值矩阵或符号矩阵
[V,J]=jordan(A) % 除了计算矩阵A的约当标准型J，还返回相应的变换矩阵V
>> A=sym([1 2 -3; 1 2 5; 2 4 -5]);
>> [V,J]=jordan(A)
V =
[-2, 30^(1/2)/58 + 14/29, 14/29 - 30^(1/2)/58]
[ 1, 22/29 - (15*30^(1/2))/58, (15*30^(1/2))/58 + 22/29]
[ 0, 1, 1]
J =
[0, 0, 0]
[0, - 30^(1/2) - 1, 0]
[0, 0, 30^(1/2) - 1]
9.3.5 奇异值分解
S=svd(A) % 给出符号矩阵奇异值对角矩阵，其计算精度由digits函数指定
[U,S,V]=svd(A) % 输出参数U和V是两个正交矩阵，它们满足关系式A=USV'
>> rng default % 设置种子数，方便复现
>> X=rand(6) % 生成6*6随机矩阵
X =
0.8147 0.2785 0.9572 0.7922 0.6787 0.7060
0.9058 0.5469 0.4854 0.9595 0.7577 0.0318
0.1270 0.9575 0.8003 0.6557 0.7431 0.2769
0.9134 0.9649 0.1419 0.0357 0.3922 0.0462
0.6324 0.1576 0.4218 0.8491 0.6555 0.0971
0.0975 0.9706 0.9157 0.9340 0.1712 0.8235
>> X=sym(X); % 将数值矩阵转换成符号矩阵
>> digits(12) % 指定输出精度
>> S=svd(vpa(X))
S =
3.56929388263
1.22304661566
1.01009041207
0.616635007956
0.406220375211
0.0390212779873
9.4 符号方程求解
9.4.1 代数方程的求解
S=solve(eqn,var) % 求方程eqn的解，自变量由var指定
S=solve(eqn,var,Name,Value) % 使用由一个或多个名称—值参数对指定的其他选项
Y=solve(eqns,vars) % 求方程组egns的解，并返回解的结构体，自变量由vars指定
Y=solve(eqns,vars,Name,Value) % 使用由一个或多个名称一值参数对指定的其他选项
[y1,...,yN]=solve(eqns,vars) % 将解分配给变量y,...,yN
>> syms a b c x
>> eqn=a*x^2+b*x+c==0;
>> S=solve(eqn)
S =
-(b + (b^2 - 4*a*c)^(1/2))/(2*a)
-(b - (b^2 - 4*a*c)^(1/2))/(2*a)
>> A=sym([1 1/4 1/6 -1; 1 1 -1 1; 1 -1/2 -1 1; -1 -1 1 1]);
>> b=sym([0; 1; 0; 2]);
>> X1=A\b; % 直接利用运算符求解
>> syms d n p q; % 利用solve求解
>> eqns=[d+n/4+p/6==q, n+d+q-p==1, q+d-n/2==p, q+p-n-d==2];
>> Y=solve(eqns)
Y =
包含以下字段的 struct:
d: 41/42
n: 2/3
p: 15/7
q: 3/2
>> syms x y u v w
>> eqn1=u*y+v*x+2*w==0;
>> eqn2=y+x-w==0;
>> eqns=[eqn1 eqn2];
>> vars=[x,y];
>> Y=solve(eqns,vars)
Y =
包含以下字段的 struct:
x: (2*w + u*w)/(u - v)
y: -(2*w + v*w)/(u - v)
9.4.2 微分方程的求解
S=dsolve(eqn) % 求解微分方程eqn，其中eqn是一个符号方程
S=dsolve(eqn,cond) % 用初始或边界条件cond求解方程eqn
S=dsolve(___,Name,Value) % 使用由一个或多个名称—值参数对指定其他选项
[y1,...,yN]=dsolve(___) % 将解分配给变量y1,...,yN
>> xSol=dsolve('Dx=a*x','x(0)=5'); % 旧版输入方式，即将被淘汰
警告: Support for character vector or string inputs will be removed in a future release. Instead, use
syms to declare variables and replace inputs such as dsolve('Dy = -3*y') with syms y(t);
dsolve(diff(y,t) == -3*y).
> 位置：dsolve (第 126 行)
>> syms x(t) a; % 新版输入方式
>> eqn=diff(x,t)==a*x;
>> cond=x(0)==5;
>> xSol(t)=dsolve(eqn,cond)
xSol(t) =
5*exp(a*t)
>> xSol=dsolve('D2x=a^2*x','x(0)=b','Dx(0)=1'); % 旧版输入方式，即将被淘汰
警告: Support for character vector or string inputs will be removed in a future release. Instead, use
syms to declare variables and replace inputs such as dsolve('Dy = -3*y') with syms y(t);
dsolve(diff(y,t) == -3*y).
> 位置：dsolve (第 126 行)
>> syms x(t) a b; % 新版输入方式
>> eqn=diff(x,t,2)==a^2*x;
>> Dx=diff(x,t);
>> cond=[x(0)==b, Dx(0)==1];
>> xSol(t)=dsolve(eqn,cond)
xSol(t) =
(exp(a*t)*(a*b + 1))/(2*a) + (exp(-a*t)*(a*b - 1))/(2*a)
>> S=dsolve('Dx=2*y,Dy=3*x'); % 旧版输入方式，即将被淘汰
警告: Support for character vector or string inputs will be removed in a future release. Instead, use
syms to declare variables and replace inputs such as dsolve('Dy = -3*y') with syms y(t);
dsolve(diff(y,t) == -3*y).
> 位置：dsolve (第 126 行)
>> syms x(t) y(t); % 新版输入方式
>> eqns=[diff(x,t)==2*y, diff(y,t)==3*x];
>> dsolve(eqns)
ans =
包含以下字段的 struct:
y: C1*exp(6^(1/2)*t) + C2*exp(-6^(1/2)*t)
x: (6^(1/2)*C1*exp(6^(1/2)*t))/3 - (6^(1/2)*C2*exp(-6^(1/2)*t))/3
>> S=dsolve('Df=2*f+3*g,Dg=f+g','f(0)=1,g(0)=0'); % 旧版输入方式，即将被淘汰
警告: Support for character vector or string inputs will be removed in a future release. Instead, use
syms to declare variables and replace inputs such as dsolve('Dy = -3*y') with syms y(t);
dsolve(diff(y,t) == -3*y).
> 位置：dsolve (第 126 行)
>> syms f(x) g(x); % 新版输入方式
>> eqns=[diff(f,x)==2*f+3*g, diff(g,x)==f+g];
>> cond=[f(0)==1, g(0)==0];
>> dsolve(eqns,cond)
ans =
包含以下字段的 struct:
g: (13^(1/2)*exp((x*(13^(1/2) + 3))/2))/13 - (13^(1/2)*exp(-(x*(13^(1/2) - 3))/2))/13
f: (13^(1/2)*exp(-(x*(13^(1/2) - 3))/2)*(13^(1/2)/2 - 1/2))/13 + (13^(1/2)*exp((x*(13^(1/2) + 3))/2)*(13^(1/2)/2 + 1/2))/13
9.5 可视化数学分析
9.5.1 图示化符号函数计算器
>> funtool
- Insert：把当前激活窗口的函数写入列表。
- Cycle：依次循环显示fxlist中的函数。
- Delete：从fxlist列表中删除激活窗口的函数。
- Reset：使计算器恢复到初始调用状态。
- Help：获得关于界面的在线提示说明。
- Demo：自动演示。
- Close：退出。
9.5.2 Taylor级数逼近分析器
>> taylortool
浙公网安备 33010602011771号