Question Write a function that performs one step of a general s-stage diagonally implicit Runge-Kutta method to a vector alued non-homogeneous ODE: addition to the usual yo, to the step size h and the right hand side f with its Jacobian with respect to the y omponent (needed for Newton's method) the user also needs to provide the two vectors a, y and the matrix haking up the Butcher tableau of the Runge-Kutta method. You can assume that has only zero entries above the iagonal since we are only looking at diagonally implicit RK methods. Function template Python def dirk(f,Df, t0,y0, h, alpha, beta,gamma): # your code to compute y return y y' (t) = f(t, y(t)) temark: I'm using le 15 as tolerance for the Newton method - the O(h²) I suggested for the BE method might ot be good enough for higher order methods. or example: -
Question Write a function that performs one step of a general s-stage diagonally implicit Runge-Kutta method to a vector alued non-homogeneous ODE: addition to the usual yo, to the step size h and the right hand side f with its Jacobian with respect to the y omponent (needed for Newton's method) the user also needs to provide the two vectors a, y and the matrix haking up the Butcher tableau of the Runge-Kutta method. You can assume that has only zero entries above the iagonal since we are only looking at diagonally implicit RK methods. Function template Python def dirk(f,Df, t0,y0, h, alpha, beta,gamma): # your code to compute y return y y' (t) = f(t, y(t)) temark: I'm using le 15 as tolerance for the Newton method - the O(h²) I suggested for the BE method might ot be good enough for higher order methods. or example: -
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
Answer the given question with a proper explanation and step-by-step solution.
for the question above, the function defined in Python failed to pass the test, could you fix it?
![Question
Write a function that performs one step of a general s-stage diagonally implicit Runge-Kutta method to a vector
valued non-homogeneous ODE:
In addition to the usual yo, to the step size h and the right hand side f with its Jacobian with respect to the y
component (needed for Newton's method) the user also needs to provide the two vectors a, y and the matrix ß
making up the Butcher tableau of the Runge-Kutta method. You can assume that has only zero entries above the
diagonal since we are only looking at diagonally implicit RK methods.
Function template
Python
def dirk(f, Df, t0,y0, h, alpha, beta, gamma):
# your code to compute y
return y
Remark: I'm using le 15 as tolerance for the Newton method - the O(h²) I suggested for the BE method might
not be good enough for higher order methods.
For example:
Test code
from numpy import array
aCN = array( [0., 1.1)
gCN= array( [0.5, 0.5])
bCN = array( [[0., 0.1, [0., 1.11)
y dirk(lambda t,y: -10*y,
lambda t,y: array([-10]),
0, array([1.]), 1./20, aCN, bCN, gCN)
end="")
print("%1.3e " % y,
In [1]: import numpy as np
def dirk(f, Df, te, ye, h, alpha, beta, gamma):
s len(alpha)
n= len(ye)
y = ye.copy()
# Newton iteration
for i in range(s):
return res
delta = np. linalg.solve (1(y), -F(y))
y += delta
def F(x):
res = x - y - h*sum (beta[i][j]*f(te+alpha[j]*h, x) for j in range(i+1))
return res
return y
y' (t) = f(t, y(t))
def 3(x):
res = np.eye(n) h*beta[i][i] *Df (to+alpha[i]*h, x)
for j in range(i):
resh beta[i][j]*Df(te+alpha[j]*h, x)
In [2]: from numpy import array
aCN= array( [0., 1.])
gCN= array( [0.5, 0.5])
bCN= array([[0., 0.],[0., 1.]])
y = dirk(lambda t,y: -10*y,
lambda t,y: array([-10]),
e, array([1.]), 1./20, aCN, bCN, gCN)
print("%1.3e" %y, end="")
Result
6.667e-01
5.833e-01](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15d7109e-469c-48b5-b6f5-906596388e95%2F95ebdd99-f61d-461a-9539-a258aaccbf9c%2Fb7prs7e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question
Write a function that performs one step of a general s-stage diagonally implicit Runge-Kutta method to a vector
valued non-homogeneous ODE:
In addition to the usual yo, to the step size h and the right hand side f with its Jacobian with respect to the y
component (needed for Newton's method) the user also needs to provide the two vectors a, y and the matrix ß
making up the Butcher tableau of the Runge-Kutta method. You can assume that has only zero entries above the
diagonal since we are only looking at diagonally implicit RK methods.
Function template
Python
def dirk(f, Df, t0,y0, h, alpha, beta, gamma):
# your code to compute y
return y
Remark: I'm using le 15 as tolerance for the Newton method - the O(h²) I suggested for the BE method might
not be good enough for higher order methods.
For example:
Test code
from numpy import array
aCN = array( [0., 1.1)
gCN= array( [0.5, 0.5])
bCN = array( [[0., 0.1, [0., 1.11)
y dirk(lambda t,y: -10*y,
lambda t,y: array([-10]),
0, array([1.]), 1./20, aCN, bCN, gCN)
end="")
print("%1.3e " % y,
In [1]: import numpy as np
def dirk(f, Df, te, ye, h, alpha, beta, gamma):
s len(alpha)
n= len(ye)
y = ye.copy()
# Newton iteration
for i in range(s):
return res
delta = np. linalg.solve (1(y), -F(y))
y += delta
def F(x):
res = x - y - h*sum (beta[i][j]*f(te+alpha[j]*h, x) for j in range(i+1))
return res
return y
y' (t) = f(t, y(t))
def 3(x):
res = np.eye(n) h*beta[i][i] *Df (to+alpha[i]*h, x)
for j in range(i):
resh beta[i][j]*Df(te+alpha[j]*h, x)
In [2]: from numpy import array
aCN= array( [0., 1.])
gCN= array( [0.5, 0.5])
bCN= array([[0., 0.],[0., 1.]])
y = dirk(lambda t,y: -10*y,
lambda t,y: array([-10]),
e, array([1.]), 1./20, aCN, bCN, gCN)
print("%1.3e" %y, end="")
Result
6.667e-01
5.833e-01
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