Solving Matrix Differential Equation in Python using Scipy/Numpy- NDSolve equivalent?
I have two numpy arrays: 9x9 and 9x1. I'd like to solve the differential equation at discrete time points, but am having trouble getting ODEInt to work. I do am unsure if I'm even doing the right thing.
With Mathematica, the equation is:
Solution = {A[t]} /. NDSolve[{A'[t] == Ab.A[t] && A[0] == A0}, {A[t]}, {t, 0, .5}, MaxSteps -> \[Infinity]];
time = 0.25;
increment = 0.05;
MA = Table[Solution, {t, 0, time, increment}];
Where Ab is the 9x9 matrix, A0 is the 9x1 matrix (initial). Here, I solve for time and life is good.
In Python implementation I have the following code which gives me the wrong answer:
from scipy.integrate import odeint
from numpy import array, dot, pi
def deriv(A, t, Ab):
return dot(Ab, A)
def MatrixBM3(k12,k21,k13,k31,k23,k32,delta1,delta2,delta3,
w1, R1, R2):
K = array([[-k21 -k23, k12, k32, 0., 0., 0., 0., 0., 0.],
[k21, -k12 - k13, k31, 0., 0., 0., 0., 0., 0.],
[k23, k13, -k31 - k32, 0., 0., 0., 0., 0., 0.],
[0., 0., 0., -k21 - k23, k12, k32, 0., 0., 0.],
[0., 0., 0., k21, -k12 - k13, k31, 0., 0., 0.],
[0., 0., 0., k23, k13, -k31 - k32, 0., 0., 0.],
[0., 0., 0., 0., 0., 0., -k21 - k23, k12, k32],
[0., 0., 0., 0., 0., 0., k21, -k12 - k13, k31],
[0., 0., 0., 0., 0., 0., k23, k13, -k31 - k32]])
Der = array([[0., 0., 0., -delta2, 0., 0., 0., 0., 0.],
[0., 0., 0., 0., -delta1, 0., 0., 0., 0.],
[0., 0., 0., 0., 0., -delta3, 0., 0., 0.],
[delta2, 0., 0., 0., 0., 0., 0., 0., 0.],
[0., delta1, 0., 0., 0., 0., 0., 0., 0.],
[0., 0., delta3, 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.]])
W = array([[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., w1, 0., 0.],
[0., 0., 0., 0., 0., 0., 0., w1, 0.],
[0., 0., 0., 0., 0., 0., 0., 0., w1],
[0., 0., 0., w1, 0., 0., 0., 0., 0.],
[0., 0., 0., 0., w1, 0., 0., 0., 0.],
[0., 0., 0., 0., 0., w1, 0., 0., 0.]])*2*pi
R = array([[-R2, 0., 0., 0., 0., 0., 0., 0., 0.],
[0., -R2, 0., 0., 0., 0., 0., 0., 0.],
[0., 0., -R2, 0., 0., 0., 0., 0., 0.],
[0., 0., 0., -R2, 0., 0., 0., 0., 0.],
[0., 0., 0., 0., -R2, 0., 0., 0., 0.],
[0., 0., 0., 0., 0., -R2, 0., 0., 0.],
[0., 0., 0., 0., 0., 0., -R1, 0., 0.],
[0., 0., 0., 0., 0., 0., 0., -R1, 0.],
[0., 0., 0., 0., 0., 0., 0., 0., -R1]])
return(K + Der + W + R)
Ab = MatrixBM3(21.218791062154633, 17653.497151475527, 40.50203461096454, 93956.36617049483, 0.0, 0.0, -646.4238856161137, 6727.748368359598, 20919.132768439955, 200.0, 2.36787, 5.39681)
A0 = array([-0.001071585381162955, -0.89153191708755708, -0.00038431516707591748, 0.0, 0.0, 0.0, 0.00054009700135979673, 0.4493470361764082, 0.00019370128872934646])
time = array([0.0,0.05,0.1,0.15,0.2,0.25])
MA = odeint(deriv, A0, time, args=(Ab,), maxsteps=2000)
Output is:
[[ -1.07158538e-003 -8.91531917e-001 -3.84315167e-004 0.00000000e+000
0.00000000e+000 0.00000000e+000 5.40097001e-004 4.49347036e-001
1.93701289e-004]
[ 3.09311322e+019 9.45061860e+022 2.35327270e+019 2.11901406e+020
1.63784238e+023 7.60569684e+019 2.29098804e+020 1.89766602e+023
8.18752241e+019]
[ 9.84409730e+042 3.00774018e+046 7.48949158e+042 6.74394343e+043
5.21257342e+046 2.42057805e+043 7.29126532e+043 6.03948436e+046
2.60574901e+043]
[ 3.13296814e+066 9.57239028e+069 2.38359473e+066 2.14631766e+067
1.65894606e+070 7.70369662e+066 2.32050753e+067 1.92211754e+070
8.29301904e+066]
[ 9.97093898e+089 3.04649506e+093 7.58599405e+089 6.83083947e+090
5.27973769e+093 2.45176732e+090 7.38521364e+090 6.11730342e+093
2.63932422e+090]
[ 3.17333659e+113 9.69573101e+116 2.41430747e+113 2.17397307e+114
1.68032166e+117 7.80295913e+113 2.35040739e+114 1.94688412e+117
8.39987500e+113]]
But the correct answer should be:
{-0.0010733126291998989, -0.8929689437405254, -0.0003849346301906338, 0., 0., 0., 0.0005366563145999495, 0.4464844718702628, 0.00019246731509531696}
{-0.000591095648651598, -0.570032546156741, -0.00023381082725213798, -0.00024790706920038567, 0.00010389803046880286, -0.00005361569187144767, 0.0003273277204077012, 0.2870035216110215, 0.00012300339326137006}
{-0.0003770535829276868, -0.364106358478121, -0.0001492324135668267, -0.0001596072774600538, -0.0011479989178276948, -0.000034744485507007025, 0.00020965172928479557, 0.18378613639965447, 0.00007876820247280559}
{-0.00024100792803807562, -0.23298939195213314, -0.00009543704274825206, -0.00010271831380730501, -0.0013205519868311284, -0.000022472380871477824, 0.00013326471695185768, 0.11685506361394844, 0.00005008078740423007}
{-0.00015437993249587976, -0.1491438843823813, -0.00006111736454518403, -0.00006545797627466387, -0.0005705018939767294, -0.000014272382451480663, 0.00008455890984798549, 0.0741820536557778, 0.00003179071165818503}
{-0.00009882799610556456, -0.09529950309336405, -0.00003909275555213336, -0.00004138741286392128, 0.00006303116741431477, -8.944610716890746*^-6, 0.00005406263888971806, 0.04743157303933772, 0.00002032674776723143}
Can anyone point me to what I may be doing wrong?
Answer
In the call to odeint, try changing tuple(array[Ab]) to (array(Ab),), or even just (Ab,). That is, use
MA = odeint(deriv, A0, time, (Ab,))
Without seeing how you defined A0 and Ab, I can't be sure that this will fix the problem, but the following variation of your code will work. I used a 3x3 array instead of 9x9.
import numpy as np
from scipy.integrate import odeint
def deriv(A, t, Ab):
return np.dot(Ab, A)
Ab = np.array([[-0.25, 0, 0],
[ 0.25, -0.2, 0],
[ 0, 0.2, -0.1]])
time = np.linspace(0, 25, 101)
A0 = np.array([10, 20, 30])
MA = odeint(deriv, A0, time, args=(Ab,))