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Zero
Running
on
Zero
| import torch | |
| # lagrange interpolation | |
| def lagrange_preint_o1(t1, v1, int_t_start, int_t_end): | |
| ''' | |
| lagrange interpolation of order 1 | |
| Args: | |
| t1: timestepx | |
| v1: value field at t1 | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| int1 = (int_t_end-int_t_start) | |
| return int1*v1, (int1/int1, ) | |
| def lagrange_preint_o2(t1, t2, v1, v2, int_t_start, int_t_end): | |
| ''' | |
| lagrange interpolation of order 2 | |
| Args: | |
| t1: timestepx | |
| t2: timestepy | |
| v1: value field at t1 | |
| v2: value field at t2 | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| int1 = 0.5/(t1-t2)*((int_t_end-t2)**2 - (int_t_start-t2)**2) | |
| int2 = 0.5/(t2-t1)*((int_t_end-t1)**2 - (int_t_start-t1)**2) | |
| int_sum = int1+int2 | |
| return int1*v1 + int2*v2, (int1/int_sum, int2/int_sum) | |
| def lagrange_preint_o3(t1, t2, t3, v1, v2, v3, int_t_start, int_t_end): | |
| ''' | |
| lagrange interpolation of order 3 | |
| Args: | |
| t1: timestepx | |
| t2: timestepy | |
| t3: timestepz | |
| v1: value field at t1 | |
| v2: value field at t2 | |
| v3: value field at t3 | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| int1_denom = (t1-t2)*(t1-t3) | |
| int1_end = 1/3*(int_t_end)**3 - 1/2*(t2+t3)*(int_t_end)**2 + (t2*t3)*int_t_end | |
| int1_start = 1/3*(int_t_start)**3 - 1/2*(t2+t3)*(int_t_start)**2 + (t2*t3)*int_t_start | |
| int1 = (int1_end - int1_start)/int1_denom | |
| int2_denom = (t2-t1)*(t2-t3) | |
| int2_end = 1/3*(int_t_end)**3 - 1/2*(t1+t3)*(int_t_end)**2 + (t1*t3)*int_t_end | |
| int2_start = 1/3*(int_t_start)**3 - 1/2*(t1+t3)*(int_t_start)**2 + (t1*t3)*int_t_start | |
| int2 = (int2_end - int2_start)/int2_denom | |
| int3_denom = (t3-t1)*(t3-t2) | |
| int3_end = 1/3*(int_t_end)**3 - 1/2*(t1+t2)*(int_t_end)**2 + (t1*t2)*int_t_end | |
| int3_start = 1/3*(int_t_start)**3 - 1/2*(t1+t2)*(int_t_start)**2 + (t1*t2)*int_t_start | |
| int3 = (int3_end - int3_start)/int3_denom | |
| int_sum = int1+int2+int3 | |
| return int1*v1 + int2*v2 + int3*v3, (int1/int_sum, int2/int_sum, int3/int_sum) | |
| def larange_preint_o4(t1, t2, t3, t4, v1, v2, v3, v4, int_t_start, int_t_end): | |
| ''' | |
| lagrange interpolation of order 4 | |
| Args: | |
| t1: timestepx | |
| t2: timestepy | |
| t3: timestepz | |
| t4: timestepw | |
| v1: value field at t1 | |
| v2: value field at t2 | |
| v3: value field at t3 | |
| v4: value field at t4 | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| int1_denom = (t1-t2)*(t1-t3)*(t1-t4) | |
| int1_end = 1/4*(int_t_end)**4 - 1/3*(t2+t3+t4)*(int_t_end)**3 + 1/2*(t3*t4 + t2*t3 + t2*t4)*int_t_end**2 - t2*t3*t4*int_t_end | |
| int1_start = 1/4*(int_t_start)**4 - 1/3*(t2+t3+t4)*(int_t_start)**3 + 1/2*(t3*t4 + t2*t3 + t2*t4)*int_t_start**2 - t2*t3*t4*int_t_start | |
| int1 = (int1_end - int1_start)/int1_denom | |
| int2_denom = (t2-t1)*(t2-t3)*(t2-t4) | |
| int2_end = 1/4*(int_t_end)**4 - 1/3*(t1+t3+t4)*(int_t_end)**3 + 1/2*(t3*t4 + t1*t3 + t1*t4)*int_t_end**2 - t1*t3*t4*int_t_end | |
| int2_start = 1/4*(int_t_start)**4 - 1/3*(t1+t3+t4)*(int_t_start)**3 + 1/2*(t3*t4 + t1*t3 + t1*t4)*int_t_start**2 - t1*t3*t4*int_t_start | |
| int2 = (int2_end - int2_start)/int2_denom | |
| int3_denom = (t3-t1)*(t3-t2)*(t3-t4) | |
| int3_end = 1/4*(int_t_end)**4 - 1/3*(t1+t2+t4)*(int_t_end)**3 + 1/2*(t4*t2 + t1*t2 + t1*t4)*int_t_end**2 - t1*t2*t4*int_t_end | |
| int3_start = 1/4*(int_t_start)**4 - 1/3*(t1+t2+t4)*(int_t_start)**3 + 1/2*(t4*t2 + t1*t2 + t1*t4)*int_t_start**2 - t1*t2*t4*int_t_start | |
| int3 = (int3_end - int3_start)/int3_denom | |
| int4_denom = (t4-t1)*(t4-t2)*(t4-t3) | |
| int4_end = 1/4*(int_t_end)**4 - 1/3*(t1+t2+t3)*(int_t_end)**3 + 1/2*(t3*t2 + t1*t2 + t1*t3)*int_t_end**2 - t1*t2*t3*int_t_end | |
| int4_start = 1/4*(int_t_start)**4 - 1/3*(t1+t2+t3)*(int_t_start)**3 + 1/2*(t3*t2 + t1*t2 + t1*t3)*int_t_start**2 - t1*t2*t3*int_t_start | |
| int4 = (int4_end - int4_start)/int4_denom | |
| int_sum = int1+int2+int3+int4 | |
| return int1*v1 + int2*v2 + int3*v3 + int4*v4, (int1/int_sum, int2/int_sum, int3/int_sum, int4/int_sum) | |
| def lagrange_preint(order, pre_vs, pre_ts, int_t_start, int_t_end): | |
| ''' | |
| lagrange interpolation | |
| Args: | |
| order: order of interpolation | |
| pre_vs: value field at pre_ts | |
| pre_ts: timesteps | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| order = min(order, len(pre_vs), len(pre_ts)) | |
| if order == 1: | |
| return lagrange_preint_o1(pre_ts[-1], pre_vs[-1], int_t_start, int_t_end) | |
| elif order == 2: | |
| return lagrange_preint_o2(pre_ts[-2], pre_ts[-1], pre_vs[-2], pre_vs[-1], int_t_start, int_t_end) | |
| elif order == 3: | |
| return lagrange_preint_o3(pre_ts[-3], pre_ts[-2], pre_ts[-1], pre_vs[-3], pre_vs[-2], pre_vs[-1], int_t_start, int_t_end) | |
| elif order == 4: | |
| return larange_preint_o4(pre_ts[-4], pre_ts[-3], pre_ts[-2], pre_ts[-1], pre_vs[-4], pre_vs[-3], pre_vs[-2], pre_vs[-1], int_t_start, int_t_end) | |
| else: | |
| raise ValueError('Invalid order') | |
| def polynomial_integral(coeffs, int_t_start, int_t_end): | |
| ''' | |
| polynomial integral | |
| Args: | |
| coeffs: coefficients of the polynomial | |
| int_t_start: intergation start time | |
| int_t_end: intergation end time | |
| Returns: | |
| integrated value | |
| ''' | |
| orders = len(coeffs) | |
| int_val = 0 | |
| for o in range(orders): | |
| int_val += coeffs[o]/(o+1)*(int_t_end**(o+1)-int_t_start**(o+1)) | |
| return int_val | |