f2:0.0;
eps:read("Please, enter epsilon: "); print("Epsilon = ", eps);
while (f1 - f2) >= eps do( f2 : f1,
i : i + 1.0,
f1 : f1 + ((3.0*i + 1)^2)/((4.0*i*i - 1.0)^2.0) );
print("Sum of series = ", f1);
Рис. 10. Результат задания 2.
Метод сопряженных градиентов
|
æ 3 |
4 |
0ö |
æ |
1ö |
|||
Решить систему Ax=B, где |
ç |
4 |
- 3 |
0 |
÷ |
ç |
5 |
÷ |
A = ç |
÷ |
B = ç |
÷ |
|||||
|
ç |
0 |
0 |
5 |
÷ |
ç |
9 |
÷ |
|
è |
ø |
è |
ø |
||||
Файл soprgrad.mac:
SoprGrad(A, n) :=block([y,x,Ax,Ap,r,r1,p,epsilon, k:0, atemp:0.0, btemp:0.0],
print("< Solution by conjugate gradients >"), print("Enter initial approximation for matrix:"), for i:0 thru n-1 do(
print("X[",i+1,"] = "), x[i]:read()
),
epsilon:read("Enter precision:"), if epsilon <= 0 then block(
print("Error! Precision must be more then 0"), break
),
Ax:MatrixByVector(A, x, n), for i:0 thru n-1 do(
r[i] : A[i, n] - Ax[i]
),
if Scal(r, r, n) # 0 then block(
for i:0 thru n-1 do p[i] : r[i], do(
Ap : MatrixByVector(A, p, n), atemp : Scal(r, p, n) / Scal(p, Ap, n), for i:0 thru n-1 do(
y[i] : x[i] + atemp * p[i], r1[i] : r[i] - atemp * Ap[i]
),
if NormaV(r1, n) <= epsilon then joe(
print("Amount of iterations: ", k), return (y)
),
btemp : Scal(r1, Ap, n) / Scal(p, Ap, n), for i:0 thru n-1 do(
p[i] : r1[i] - btemp * p[i], r[i] : r1[i],
x[i] : y[i]
),
k : k + 1
)
)
else block(
print("Amount of iterations: 1"), return (x)
)
);
MatrixByVector( H, V, n) := block([tmp,summa:0.0], for i:0 thru n-1 do(
for j:0 thru n-1 do
summa : summa + V[j] * H[i,j], tmp[i] : summa,
summa : 0.0
),
return (tmp)
);
NormaV(V, n):= block([summa:0.0],
for i:0 thru n-1 do summa : summa + V[i]*V[i], summa : sqrt(summa),
return (summa)
);
Scal(v1, v2, n):= block([summa:0.0], for i : 0 thru n-1 do
summa : summa + v1[i] * v2[i], return (summa)
);
n : 3;
array(A, n-1, n); for i:0 thru n-1 do(
for j:0 thru n do(
print("Enter A[", i, ",", j, "]: "), A[i,j]:read()
) |
|
|
); |
|
|
print("Coefficients of extended matrix:"); |
|
|
for i:0 thru n-1 do( |
|
|
print(A[i, 0], "", A[i, 1], " |
",A[i, 2]," |
",A[i, 3]) |
);
temp : SoprGrad(A, n); for i:0 thru n-1 do
print("X[", i + 1, "] = ", temp[i]);
Рис. 11. Результат задания метода сопряженных градиентов.
Решение нелинейных уравнений
Требуется решить методом простой итерации нелинейное уравнение x2 - 11× x + 30 = 0 , интервал поиска корня [3, 5.5]и шаг h = 0.3 .
Требуется решить уравнение f (x) = 0 с точностью ε: f (x) = x3 + 2x2 + 3x + 5,
ε = 0.00001.
Файл nonlinear.mac:
coefficients(coefs, n) := block( [prived_coefs],
if coefs[1] = 0 then break("incorrect equation!!!"), for i : 0 thru n-1 do prived_coefs[i]:-coefs[i]/coefs[1], prived_coefs[1] : 0.0,
return (prived_coefs)
);
f(x, coefs, n):= block( [result:0.0],
for i:0 thru n-1 do result : result + (x^i)*coefs[i], return (result)
);
StepsMeth(st, coefs, n, a, b) := block( [x:0.0, x_prev:0.0, b_new:b], for i:a thru b-st step st do(
x_prev : x,
x : f(i, coefs, n),
if x*x_prev < 0 then joe( b_new : i, return()
), |
|
if |
x = 0 then joe( |
b_new : i+st/2, return ()
)
),
return (b_new)
);
dir( x, coefs, n) := block( [result:0.0],
for i:1 thru n-1 do result : result + x^(i - 1)*i*coefs[i], return (result)
);
dirSecond(x, coefs, n) := block( [result:0.0],
for i:2 thru n-1 do result : result + x^(i - 2)*(i)*(i-1)*coefs[i], return (result)
);
halfDivision(coefs, n, st, at, bt, e) := block( [b:at, x:at, a:0.0],
do(
b : StepsMeth(st, coefs, n , b, bt), a : b - st,
if bt <= b then return(), x : a,
while (abs(f(x, coefs, n)) > e) do( x : (b+a)/2,
if f(x,coefs,n)*f(a,coefs,n) < 0 then b : x else a : x,
if f(x,coefs,n) = 0 then joe( print("Root: ", x), return()
)
),
print("Root: ", x)
)
);
Xord(coefs, n, st, at, bt, e) := block( [b:at, x:at, a:0.0],
do(
b : StepsMeth(st, coefs, n, b, bt), a : b - st,
if bt <= b then return(), x : a,
while (abs(f(x, coefs, n)) > e) do
x : -(f(x,coefs,n)*(b-x))/(f(b,coefs,n)-f(x,coefs,n)) + x, print("Root: ", x)
)
);
Casatel(coefs, n, st, at, bt, e) := block( [b:at, x:at, a:0.0],
do(
b : StepsMeth(st, coefs, n , b, bt), a : b - st,
if bt <= b then return(),
if dir(a, coefs,n) > 0 then x : b else x : a,
while (abs(f(x,coefs,n)) > e) do
x : -f(x,coefs,n)/dir(x, coefs, n) + x, print("Root: ", x)
)
);
Iteration(coefs, n, st, at, bt, e) := block( [b:at, x:at, a:0.0, prived_coefs], do(
b : StepsMeth(st, coefs, n , b, bt), a : b - st,
if bt <= b then return(), x : a,
prived_coefs : coefficients(coefs, n), while(abs(f(x,coefs,n)) > e) do x : f(x, prived_coefs ,n), print("Root: ", x)
)
);
Graph(coefs, n, at, bt):= block([f:0.0],
for i:0 thru n-1 do f : f + (x^i)*coefs[i], plot2d(f,[x, at, bt])
);
print("<Solution of nonlinear equation>"); n : read("Enter power of polynomial: "); array(coefs, n-1);
for i:0 thru n-1 do(
print("Enter ",i,"-th coefficient: "), coefs[i]: read()
);
st : 0.1;
at : read("Enter a: "); bt : read("Enter b: ");
e : read("Enter presicion: "); print("Chose method:"); print("1 - halfDivision,"); print("2 - Casatelnih,"); print("3 - Hord,");
print("4 - Iteracii,");
print("5 - graphical method."); method : read();
if method = 1 then halfDivision(coefs, n, st, at, bt, e); if method = 2 then Casatel(coefs, n, st, at, bt, e);
if method = 3 then Xord(coefs, n, st, at, bt, e);
if method = 4 then Iteration(coefs, n, st, at, bt, e); if method = 5 then Graph(coefs, n, at, bt);
Рис. 12. Метод деления отрезка пополам.
Рис. 13. Графический.
Рис. 14. Метод Хорд.
Рис. 15. Метод касательных (Ньютона).