北邮计算物理实践ch3数组作业
北邮计算物理实践ch3数组作业
1.
d=pi/100;
t=0:d:2*pi;
x=sin(t);
dt=10*eps; %
x_eps=sin(t+dt);
dxdt_eps=(x_eps-x)/dt; %
plot(t,x,'LineWidth',5)
hold on
plot(t,dxdt_eps)
hold off
legend('x(t)','dx/dt')
xlabel('t')
t1n = ndims(t1)
t1s = size(t1)
t1l = length(t1)
Tn = ndims(T)
Ts = size(T)
Tl = length(T)
Fn = ndims(F)
Fs = size(F)
Fl =length(F)
T12 = (t1==t2)
T12 =
1×10 logical 数组
1 1 1 1 1 1 0 1 1 1
T12(7)=0,说明t1在第七个元素处不等于t2，我们可以来看一下：
format long
t1_7 = t1(7)
t1_7 =
2.094395102393196
t2_7 = t2(7)
t2_7 =
2.094395102393195
可以看出t1与t2的第七个元素在最后一位不同
2.
A = reshape(1:18,3,6)
A =
1 4 7 10 13 16
2 5 8 11 14 17
3 6 9 12 15 18
A([2,4,8,16]) = NaN
A =
1 NaN 7 10 13 NaN
NaN 5 NaN 11 14 17
3 6 9 12 15 18
A(:,[4,5]) = Inf
A =
1 NaN 7 Inf Inf NaN
NaN 5 NaN Inf Inf 17
3 6 9 Inf Inf 18
3.
rng('default')
A = rand(3,5)
A =
0.8147 0.9134 0.2785 0.9649 0.9572
0.9058 0.6324 0.5469 0.1576 0.4854
0.1270 0.0975 0.9575 0.9706 0.8003
[ar,aj] = find(A>0.5);
display(['全下标索引的行号',int2str(ar')])
display(['全下标索引的列号',int2str(aj')])
全下标索引的行号1 2 1 2 2 3 1 3 1 3
全下标索引的列号1 1 2 2 3 3 4 4 5 5
ak = find(A>0.5);
display(['单下标',int2str(ak')])
单下标1 2 4 5 8 9 10 12 13 15
4.
fprintf('标量循环法的纵轴是y1,数组运算法是y2');
A = linspace(0,10,100);
N = length(A);
B = zeros(size(A));
for k = 1:N
B(k) = 1-exp(-0.5*A(k))*cos(2*A(k));
end
subplot(1,2,1)
plot(A,B)
xlabel('t');
ylabel('y1');
grid on
C = 1-exp(-0.5*A).*cos(2*A);
subplot(1,2,2)
plot(A,C)
xlabel('t');
ylabel('y2');
grid on
标量循环法的纵轴是y1,数组运算法是y2
5.
A = magic(3); B = rand(3);
(1)
AB1 = A.*B == B.*A
AB1 =
3×3 logical 数组
1 1 1
1 1 1
1 1 1
相同
(2)
AB2 = A*B == A.*B
AB2 =
3×3 logical 数组
0 0 0
0 0 0
0 0 0
不相同，因为A*B是正常代数上的矩阵乘法，而点乘是对应元素相乘
(3)
AB3 = A*B == B*A
AB3 =
3×3 logical 数组
0 0 0
0 0 0
0 0 0
不同，显然代数中的A*B矩阵和B*A矩阵不同
(4)
AB4 = A.\B == B./A
AB4 =
3×3 logical 数组
1 1 1
1 1 1
1 1 1
相同，因为点除都是B矩阵元素除以A矩阵对应元素
(5)
AB5 = A\B == B/A
AB5 =
3×3 logical 数组
0 0 0
0 0 0
0 0 0
不同，因为A\B是Ax = B中的解x，而B/A是xA = B中的解，在第(3)题中已经论证Ax与xA结果不同
(6)
AB6 = A*A\B-B == A*(A\B)-B
AB6 =
3×3 logical 数组
0 0 0
0 0 0
0 0 0
不同，第二个接近于0
(7)
AB7 = A\eye(3) == eye(3)/A
AB7 =
3×3 logical 数组
1 1 0
0 1 1
0 1 0
运算结果相同，因为得到的都是A的逆矩阵
6.
A = [1 2; 3 4]
A =
1 2
3 4
(1)
B1 = A.^0.5
B1 =
1.0000 1.4142
1.7321 2.0000
B2 = 0.5.^A
B2 =
0.5000 0.2500
0.1250 0.0625
B3 = A^0.5
B3 =
0.5537 + 0.4644i 0.8070 - 0.2124i
1.2104 - 0.3186i 1.7641 + 0.1458i
B4 = 0.5^A
B4 =
0.9910 -0.4422
-0.6634 0.3276
(2)
A1 = B1.^2;
A2 = log(B2)/log(0.5);
A3 = B3^2;
A4 = logm(B4)/logm(0.5);
(3)
其实这些矩阵实部都一样，虚部为0而已7.
x = -3*pi:pi/15:3*pi;
y = x;
[X,Y] = meshgrid(x,y);
warning off;
Z = sin(X).*sin(Y)./X./Y;
(1)
Z中有91+90 = 181 个非数数据，
L = find(isnan(Z)); %NaN ~= NaN,所以不能用find(Z == NaN)
n = length(L)
n =
181
surf(X,Y,Z)
shading interp
(2)
x = -3*pi:pi/15:3*pi;
y = x;
[X,Y] = meshgrid(x,y);
X2 = X + find(X == 0)*realmin;
Y2 = Y + find(Y == 0)*realmin;
warning off;
Z2 = sin(X2).*sin(Y2)./X2./Y2;
surf(X2,Y2,Z2)
shading interp
8.
(1)标量循环+条件分支法：
x = -1.5:0.001:1.5; y = -3:0.001:3;
[X,Y] = meshgrid(x,y);
xl = length(x); yl = length(y);
z = zeros(xl,yl);
for ii = 1:xl
for jj = 1:yl
if ( x(ii)+y(jj)<=-1)
z(ii,jj) = 0.546*exp(-0.75*y(jj)^2-3.75*x(ii)^2+1.5*x(ii));
elseif (x(ii)+y(jj)>-1 & x(ii)+y(jj)<=1)
z(ii,jj) = 0.758*exp(-y(jj)^2-6*x(ii)^2);
else
z(ii,jj) = 0.546*exp(-0.75*y(jj)^2-3.75*x(ii)^2-1.5*x(ii));
end
end
end
surf(X,Y,z')
shading interp
(2)数组混合
x = -1.5:0.001:1.5; y = -3:0.001:3;
[X,Y] = meshgrid(x,y);
Z1 = 0.546*exp(-0.75*Y.^2-3.75*X.^2+1.5*X).*(X+Y<=-1);
Z2 = 0.758*exp(-Y.^2-6*X.^2).*(X+Y>-1 & X+Y<=1);
Z3 = 0.546*exp(-0.75*Y.^2-3.75*X.^2-1.5*X).*(X+Y>1);
Z = Z1+Z2+Z3;
surf(X,Y,Z)
shading interp
(3)比较两种方法的元素相对误差
format long
m1 = abs(z'-Z);
m2 = max(m1(:))
9.
(1)
function D = matrix_old_multiply(A,B)
% 矩阵的乘法
% 对应行列相乘相加
[a,b] = size(A);
[c,d] = size(B);
if b~=c
error('A矩阵的列数不等于B矩阵的行数，不能相乘啦！')
end
for ii = 1:a
for jj = 1:d
new_i = 0; new_j = 0;
for k = 1:b
new_i = new_i + real(A(ii,k))*real(B(k,jj))...
-imag(A(ii,k))*imag(B(k,jj));
new_j = new_j + real(A(ii,k))*imag(B(k,jj))...
+imag(A(ii,k))*real(B(k,jj));
end
D(ii,jj) = new_i + i*new_j;
end
end
end
计算G = (AB-C)*D只需调用两次上面的函数即可
rng default
A=randn(50,70)+1i*randn(50,70);
B=randn(70,60)+1i*randn(70,60);
C=randn(50,60)+1i*randn(50,60);
D=randn(60,1)+1i*randn(60,1);
K = matrix_old_multiply(A,B);
G = matrix_old_multiply(K-C,D);
(2)
G2 = (A*B-C)*D;
N1 = norm(real(G2))
N2 = norm(imag(G2))
N1 =
6.352169123696923e+02
N2 =
8.048619385149246e+02