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The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-1]) + f(x[i+1]) =
2 * f'(x[i]) * h + 1/3 * f'''(x[i]) * h^3 + 1/60 * f^(5)(x[i]) * h^5 + 1/2520 * f^(7)(x[i]) * h^7 + 1/181440 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f'(x[i]) = ( -f(x[i-1]) + f(x[i+1]) ) / (2 * h) + O(h^2)

Or

f'(x[i]) = ( -1/2 * f(x[i-1]) + 1/2 * f(x[i+1]) ) / h + O(h^2)

Julia function:

f1stderiv2ptcentrale(f, x, i, h) = ( -f(x[i-1]) + f(x[i+1]) ) / (2 * h)

Or

f1stderiv2ptcentrale1(f, x, i, h) = ( -1/2 * f(x[i-1]) + 1/2 * f(x[i+1]) ) / h

Or

f1stderiv2ptcentrald(f, x, i, h) = ( -0.500000 * f(x[i-1]) + 0.500000 * f(x[i+1]) ) / h

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-2]) - 8 * f(x[i-1]) + 8 * f(x[i+1]) - f(x[i+2]) =
12 * f'(x[i]) * h - 2/5 * f^(5)(x[i]) * h^5 - 1/21 * f^(7)(x[i]) * h^7 - 1/360 * f^(9)(x[i]) * h^9 - 17/166320 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f'(x[i]) = ( f(x[i-2]) - 8 * f(x[i-1]) + 8 * f(x[i+1]) - f(x[i+2]) ) / (12 * h) + O(h^4)

Or

f'(x[i]) = ( 1/12 * f(x[i-2]) - 2/3 * f(x[i-1]) + 2/3 * f(x[i+1]) - 1/12 * f(x[i+2]) ) / h + O(h^4)

Julia function:

f1stderiv4ptcentrale(f, x, i, h) = ( f(x[i-2]) - 8 * f(x[i-1]) + 8 * f(x[i+1]) - f(x[i+2]) ) / (12 * h)

Or

f1stderiv4ptcentrale1(f, x, i, h) = ( 1/12 * f(x[i-2]) - 2/3 * f(x[i-1]) + 2/3 * f(x[i+1]) - 1/12 * f(x[i+2]) ) / h

Or

f1stderiv4ptcentrald(f, x, i, h) = ( 0.083333 * f(x[i-2]) - 0.666667 * f(x[i-1]) + 0.666667 * f(x[i+1]) - 0.083333 * f(x[i+2]) ) / h

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-3]) + 9 * f(x[i-2]) - 45 * f(x[i-1]) + 45 * f(x[i+1]) - 9 * f(x[i+2]) + f(x[i+3]) =
60 * f'(x[i]) * h + 3/7 * f^(7)(x[i]) * h^7 + 1/12 * f^(9)(x[i]) * h^9 + 7/880 * f^(11)(x[i]) * h^11 + 2/4095 * f^(13)(x[i]) * h^13 + ...

The exact formula:

f'(x[i]) = ( -f(x[i-3]) + 9 * f(x[i-2]) - 45 * f(x[i-1]) + 45 * f(x[i+1]) - 9 * f(x[i+2]) + f(x[i+3]) ) / (60 * h) + O(h^6)

Or

f'(x[i]) = ( -1/60 * f(x[i-3]) + 3/20 * f(x[i-2]) - 3/4 * f(x[i-1]) + 3/4 * f(x[i+1]) - 3/20 * f(x[i+2]) + 1/60 * f(x[i+3]) ) / h + O(h^6)

Julia function:

f1stderiv6ptcentrale(f, x, i, h) = ( -f(x[i-3]) + 9 * f(x[i-2]) - 45 * f(x[i-1]) + 45 * f(x[i+1]) - 9 * f(x[i+2]) + f(x[i+3]) ) / (60 * h)

Or

f1stderiv6ptcentrale1(f, x, i, h) = ( -1/60 * f(x[i-3]) + 3/20 * f(x[i-2]) - 3/4 * f(x[i-1]) + 3/4 * f(x[i+1]) - 3/20 * f(x[i+2]) + 1/60 * f(x[i+3]) ) / h

Or

f1stderiv6ptcentrald(f, x, i, h) = ( -0.016667 * f(x[i-3]) + 0.150000 * f(x[i-2]) - 0.750000 * f(x[i-1]) + 0.750000 * f(x[i+1]) - 0.150000 * f(x[i+2]) + 0.016667 * f(x[i+3]) ) / h

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

3 * f(x[i-4]) - 32 * f(x[i-3]) + 168 * f(x[i-2]) - 672 * f(x[i-1]) + 672 * f(x[i+1]) - 168 * f(x[i+2]) + 32 * f(x[i+3]) - 3 * f(x[i+4]) =
840 * f'(x[i]) * h - 4/3 * f^(9)(x[i]) * h^9 - 4/11 * f^(11)(x[i]) * h^11 - 19/390 * f^(13)(x[i]) * h^13 - 4/945 * f^(15)(x[i]) * h^15 + ...

The exact formula:

f'(x[i]) = ( 3 * f(x[i-4]) - 32 * f(x[i-3]) + 168 * f(x[i-2]) - 672 * f(x[i-1]) + 672 * f(x[i+1]) - 168 * f(x[i+2]) + 32 * f(x[i+3]) - 3 * f(x[i+4]) ) / (840 * h) + O(h^8)

Or

f'(x[i]) = ( 1/280 * f(x[i-4]) - 4/105 * f(x[i-3]) + 1/5 * f(x[i-2]) - 4/5 * f(x[i-1]) + 4/5 * f(x[i+1]) - 1/5 * f(x[i+2]) + 4/105 * f(x[i+3]) - 1/280 * f(x[i+4]) ) / h + O(h^8)

Julia function:

f1stderiv8ptcentrale(f, x, i, h) = ( 3 * f(x[i-4]) - 32 * f(x[i-3]) + 168 * f(x[i-2]) - 672 * f(x[i-1]) + 672 * f(x[i+1]) - 168 * f(x[i+2]) + 32 * f(x[i+3]) - 3 * f(x[i+4]) ) / (840 * h)

Or

f1stderiv8ptcentrale1(f, x, i, h) = ( 1/280 * f(x[i-4]) - 4/105 * f(x[i-3]) + 1/5 * f(x[i-2]) - 4/5 * f(x[i-1]) + 4/5 * f(x[i+1]) - 1/5 * f(x[i+2]) + 4/105 * f(x[i+3]) - 1/280 * f(x[i+4]) ) / h

Or

f1stderiv8ptcentrald(f, x, i, h) = ( 0.003571 * f(x[i-4]) - 0.038095 * f(x[i-3]) + 0.200000 * f(x[i-2]) - 0.800000 * f(x[i-1]) + 0.800000 * f(x[i+1]) - 0.200000 * f(x[i+2]) + 0.038095 * f(x[i+3]) - 0.003571 * f(x[i+4]) ) / h

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-1]) - 2 * f(x[i]) + f(x[i+1]) =
f''(x[i]) * h^2 + 1/12 * f^(4)(x[i]) * h^4 + 1/360 * f^(6)(x[i]) * h^6 + 1/20160 * f^(8)(x[i]) * h^8 + 1/1814400 * f^(10)(x[i]) * h^10 + ...

The exact formula:

f''(x[i]) = ( f(x[i-1]) - 2 * f(x[i]) + f(x[i+1]) ) / h^2 + O(h^2)

Julia function:

f2ndderiv3ptcentral(f, x, i, h) = ( f(x[i-1]) - 2 * f(x[i]) + f(x[i+1]) ) / h^2

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-2]) + 16 * f(x[i-1]) - 30 * f(x[i]) + 16 * f(x[i+1]) - f(x[i+2]) =
12 * f''(x[i]) * h^2 - 2/15 * f^(6)(x[i]) * h^6 - 1/84 * f^(8)(x[i]) * h^8 - 1/1800 * f^(10)(x[i]) * h^10 - 17/997920 * f^(12)(x[i]) * h^12 + ...

The exact formula:

f''(x[i]) = ( -f(x[i-2]) + 16 * f(x[i-1]) - 30 * f(x[i]) + 16 * f(x[i+1]) - f(x[i+2]) ) / (12 * h^2) + O(h^4)

Or

f''(x[i]) = ( -1/12 * f(x[i-2]) + 4/3 * f(x[i-1]) - 5/2 * f(x[i]) + 4/3 * f(x[i+1]) - 1/12 * f(x[i+2]) ) / h^2 + O(h^4)

Julia function:

f2ndderiv5ptcentrale(f, x, i, h) = ( -f(x[i-2]) + 16 * f(x[i-1]) - 30 * f(x[i]) + 16 * f(x[i+1]) - f(x[i+2]) ) / (12 * h^2)

Or

f2ndderiv5ptcentrale1(f, x, i, h) = ( -1/12 * f(x[i-2]) + 4/3 * f(x[i-1]) - 5/2 * f(x[i]) + 4/3 * f(x[i+1]) - 1/12 * f(x[i+2]) ) / h^2

Or

f2ndderiv5ptcentrald(f, x, i, h) = ( -0.083333 * f(x[i-2]) + 1.333333 * f(x[i-1]) - 2.500000 * f(x[i]) + 1.333333 * f(x[i+1]) - 0.083333 * f(x[i+2]) ) / h^2

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

2 * f(x[i-3]) - 27 * f(x[i-2]) + 270 * f(x[i-1]) - 490 * f(x[i]) + 270 * f(x[i+1]) - 27 * f(x[i+2]) + 2 * f(x[i+3]) =
180 * f''(x[i]) * h^2 + 9/28 * f^(8)(x[i]) * h^8 + 1/20 * f^(10)(x[i]) * h^10 + 7/1760 * f^(12)(x[i]) * h^12 + 2/9555 * f^(14)(x[i]) * h^14 + ...

The exact formula:

f''(x[i]) = ( 2 * f(x[i-3]) - 27 * f(x[i-2]) + 270 * f(x[i-1]) - 490 * f(x[i]) + 270 * f(x[i+1]) - 27 * f(x[i+2]) + 2 * f(x[i+3]) ) / (180 * h^2) + O(h^6)

Or

f''(x[i]) = ( 1/90 * f(x[i-3]) - 3/20 * f(x[i-2]) + 3/2 * f(x[i-1]) - 49/18 * f(x[i]) + 3/2 * f(x[i+1]) - 3/20 * f(x[i+2]) + 1/90 * f(x[i+3]) ) / h^2 + O(h^6)

Julia function:

f2ndderiv7ptcentrale(f, x, i, h) = ( 2 * f(x[i-3]) - 27 * f(x[i-2]) + 270 * f(x[i-1]) - 490 * f(x[i]) + 270 * f(x[i+1]) - 27 * f(x[i+2]) + 2 * f(x[i+3]) ) / (180 * h^2)

Or

f2ndderiv7ptcentrale1(f, x, i, h) = ( 1/90 * f(x[i-3]) - 3/20 * f(x[i-2]) + 3/2 * f(x[i-1]) - 49/18 * f(x[i]) + 3/2 * f(x[i+1]) - 3/20 * f(x[i+2]) + 1/90 * f(x[i+3]) ) / h^2

Or

f2ndderiv7ptcentrald(f, x, i, h) = ( 0.011111 * f(x[i-3]) - 0.150000 * f(x[i-2]) + 1.500000 * f(x[i-1]) - 2.722222 * f(x[i]) + 1.500000 * f(x[i+1]) - 0.150000 * f(x[i+2]) + 0.011111 * f(x[i+3]) ) / h^2

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-9 * f(x[i-4]) + 128 * f(x[i-3]) - 1008 * f(x[i-2]) + 8064 * f(x[i-1]) - 14350 * f(x[i]) + 8064 * f(x[i+1]) - 1008 * f(x[i+2]) + 128 * f(x[i+3]) - 9 * f(x[i+4]) =
5040 * f''(x[i]) * h^2 - 8/5 * f^(10)(x[i]) * h^10 - 4/11 * f^(12)(x[i]) * h^12 - 19/455 * f^(14)(x[i]) * h^14 - 1/315 * f^(16)(x[i]) * h^16 + ...

The exact formula:

f''(x[i]) = ( -9 * f(x[i-4]) + 128 * f(x[i-3]) - 1008 * f(x[i-2]) + 8064 * f(x[i-1]) - 14350 * f(x[i]) + 8064 * f(x[i+1]) - 1008 * f(x[i+2]) + 128 * f(x[i+3]) - 9 * f(x[i+4]) ) / (5040 * h^2) + O(h^8)

Or

f''(x[i]) = ( -1/560 * f(x[i-4]) + 8/315 * f(x[i-3]) - 1/5 * f(x[i-2]) + 8/5 * f(x[i-1]) - 205/72 * f(x[i]) + 8/5 * f(x[i+1]) - 1/5 * f(x[i+2]) + 8/315 * f(x[i+3]) - 1/560 * f(x[i+4]) ) / h^2 + O(h^8)

Julia function:

f2ndderiv9ptcentrale(f, x, i, h) = ( -9 * f(x[i-4]) + 128 * f(x[i-3]) - 1008 * f(x[i-2]) + 8064 * f(x[i-1]) - 14350 * f(x[i]) + 8064 * f(x[i+1]) - 1008 * f(x[i+2]) + 128 * f(x[i+3]) - 9 * f(x[i+4]) ) / (5040 * h^2)

Or

f2ndderiv9ptcentrale1(f, x, i, h) = ( -1/560 * f(x[i-4]) + 8/315 * f(x[i-3]) - 1/5 * f(x[i-2]) + 8/5 * f(x[i-1]) - 205/72 * f(x[i]) + 8/5 * f(x[i+1]) - 1/5 * f(x[i+2]) + 8/315 * f(x[i+3]) - 1/560 * f(x[i+4]) ) / h^2

Or

f2ndderiv9ptcentrald(f, x, i, h) = ( -0.001786 * f(x[i-4]) + 0.025397 * f(x[i-3]) - 0.200000 * f(x[i-2]) + 1.600000 * f(x[i-1]) - 2.847222 * f(x[i]) + 1.600000 * f(x[i+1]) - 0.200000 * f(x[i+2]) + 0.025397 * f(x[i+3]) - 0.001786 * f(x[i+4]) ) / h^2

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-2]) + 2 * f(x[i-1]) - 2 * f(x[i+1]) + f(x[i+2]) =
2 * f'''(x[i]) * h^3 + 1/2 * f^(5)(x[i]) * h^5 + 1/20 * f^(7)(x[i]) * h^7 + 17/6048 * f^(9)(x[i]) * h^9 + 31/302400 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f'''(x[i]) = ( -f(x[i-2]) + 2 * f(x[i-1]) - 2 * f(x[i+1]) + f(x[i+2]) ) / (2 * h^3) + O(h^2)

Or

f'''(x[i]) = ( -1/2 * f(x[i-2]) + f(x[i-1]) - f(x[i+1]) + 1/2 * f(x[i+2]) ) / h^3 + O(h^2)

Julia function:

f3rdderiv4ptcentrale(f, x, i, h) = ( -f(x[i-2]) + 2 * f(x[i-1]) - 2 * f(x[i+1]) + f(x[i+2]) ) / (2 * h^3)

Or

f3rdderiv4ptcentrale1(f, x, i, h) = ( -1/2 * f(x[i-2]) + f(x[i-1]) - f(x[i+1]) + 1/2 * f(x[i+2]) ) / h^3

Or

f3rdderiv4ptcentrald(f, x, i, h) = ( -0.500000 * f(x[i-2]) + f(x[i-1]) - f(x[i+1]) + 0.500000 * f(x[i+2]) ) / h^3

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-3]) - 8 * f(x[i-2]) + 13 * f(x[i-1]) - 13 * f(x[i+1]) + 8 * f(x[i+2]) - f(x[i+3]) =
8 * f'''(x[i]) * h^3 - 7/15 * f^(7)(x[i]) * h^7 - 65/756 * f^(9)(x[i]) * h^9 - 29/3600 * f^(11)(x[i]) * h^11 - 7/14256 * f^(13)(x[i]) * h^13 + ...

The exact formula:

f'''(x[i]) = ( f(x[i-3]) - 8 * f(x[i-2]) + 13 * f(x[i-1]) - 13 * f(x[i+1]) + 8 * f(x[i+2]) - f(x[i+3]) ) / (8 * h^3) + O(h^4)

Or

f'''(x[i]) = ( 1/8 * f(x[i-3]) - f(x[i-2]) + 13/8 * f(x[i-1]) - 13/8 * f(x[i+1]) + f(x[i+2]) - 1/8 * f(x[i+3]) ) / h^3 + O(h^4)

Julia function:

f3rdderiv6ptcentrale(f, x, i, h) = ( f(x[i-3]) - 8 * f(x[i-2]) + 13 * f(x[i-1]) - 13 * f(x[i+1]) + 8 * f(x[i+2]) - f(x[i+3]) ) / (8 * h^3)

Or

f3rdderiv6ptcentrale1(f, x, i, h) = ( 1/8 * f(x[i-3]) - f(x[i-2]) + 13/8 * f(x[i-1]) - 13/8 * f(x[i+1]) + f(x[i+2]) - 1/8 * f(x[i+3]) ) / h^3

Or

f3rdderiv6ptcentrald(f, x, i, h) = ( 0.125000 * f(x[i-3]) - f(x[i-2]) + 1.625000 * f(x[i-1]) - 1.625000 * f(x[i+1]) + f(x[i+2]) - 0.125000 * f(x[i+3]) ) / h^3

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-7 * f(x[i-4]) + 72 * f(x[i-3]) - 338 * f(x[i-2]) + 488 * f(x[i-1]) - 488 * f(x[i+1]) + 338 * f(x[i+2]) - 72 * f(x[i+3]) + 7 * f(x[i+4]) =
240 * f'''(x[i]) * h^3 + 205/63 * f^(9)(x[i]) * h^9 + 13/15 * f^(11)(x[i]) * h^11 + 91/792 * f^(13)(x[i]) * h^13 + 12812/1289925 * f^(15)(x[i]) * h^15 + ...

The exact formula:

f'''(x[i]) = ( -7 * f(x[i-4]) + 72 * f(x[i-3]) - 338 * f(x[i-2]) + 488 * f(x[i-1]) - 488 * f(x[i+1]) + 338 * f(x[i+2]) - 72 * f(x[i+3]) + 7 * f(x[i+4]) ) / (240 * h^3) + O(h^6)

Or

f'''(x[i]) = ( -7/240 * f(x[i-4]) + 3/10 * f(x[i-3]) - 169/120 * f(x[i-2]) + 61/30 * f(x[i-1]) - 61/30 * f(x[i+1]) + 169/120 * f(x[i+2]) - 3/10 * f(x[i+3]) + 7/240 * f(x[i+4]) ) / h^3 + O(h^6)

Julia function:

f3rdderiv8ptcentrale(f, x, i, h) = ( -7 * f(x[i-4]) + 72 * f(x[i-3]) - 338 * f(x[i-2]) + 488 * f(x[i-1]) - 488 * f(x[i+1]) + 338 * f(x[i+2]) - 72 * f(x[i+3]) + 7 * f(x[i+4]) ) / (240 * h^3)

Or

f3rdderiv8ptcentrale1(f, x, i, h) = ( -7/240 * f(x[i-4]) + 3/10 * f(x[i-3]) - 169/120 * f(x[i-2]) + 61/30 * f(x[i-1]) - 61/30 * f(x[i+1]) + 169/120 * f(x[i+2]) - 3/10 * f(x[i+3]) + 7/240 * f(x[i+4]) ) / h^3

Or

f3rdderiv8ptcentrald(f, x, i, h) = ( -0.029167 * f(x[i-4]) + 0.300000 * f(x[i-3]) - 1.408333 * f(x[i-2]) + 2.033333 * f(x[i-1]) - 2.033333 * f(x[i+1]) + 1.408333 * f(x[i+2]) - 0.300000 * f(x[i+3]) + 0.029167 * f(x[i+4]) ) / h^3

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-2]) - 4 * f(x[i-1]) + 6 * f(x[i]) - 4 * f(x[i+1]) + f(x[i+2]) =
f^(4)(x[i]) * h^4 + 1/6 * f^(6)(x[i]) * h^6 + 1/80 * f^(8)(x[i]) * h^8 + 17/30240 * f^(10)(x[i]) * h^10 + 31/1814400 * f^(12)(x[i]) * h^12 + ...

The exact formula:

f^(4)(x[i]) = ( f(x[i-2]) - 4 * f(x[i-1]) + 6 * f(x[i]) - 4 * f(x[i+1]) + f(x[i+2]) ) / h^4 + O(h^2)

Julia function:

f4thderiv5ptcentral(f, x, i, h) = ( f(x[i-2]) - 4 * f(x[i-1]) + 6 * f(x[i]) - 4 * f(x[i+1]) + f(x[i+2]) ) / h^4

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-3]) + 12 * f(x[i-2]) - 39 * f(x[i-1]) + 56 * f(x[i]) - 39 * f(x[i+1]) + 12 * f(x[i+2]) - f(x[i+3]) =
6 * f^(4)(x[i]) * h^4 - 7/40 * f^(8)(x[i]) * h^8 - 13/504 * f^(10)(x[i]) * h^10 - 29/14400 * f^(12)(x[i]) * h^12 - 1/9504 * f^(14)(x[i]) * h^14 + ...

The exact formula:

f^(4)(x[i]) = ( -f(x[i-3]) + 12 * f(x[i-2]) - 39 * f(x[i-1]) + 56 * f(x[i]) - 39 * f(x[i+1]) + 12 * f(x[i+2]) - f(x[i+3]) ) / (6 * h^4) + O(h^4)

Or

f^(4)(x[i]) = ( -1/6 * f(x[i-3]) + 2 * f(x[i-2]) - 13/2 * f(x[i-1]) + 28/3 * f(x[i]) - 13/2 * f(x[i+1]) + 2 * f(x[i+2]) - 1/6 * f(x[i+3]) ) / h^4 + O(h^4)

Julia function:

f4thderiv7ptcentrale(f, x, i, h) = ( -f(x[i-3]) + 12 * f(x[i-2]) - 39 * f(x[i-1]) + 56 * f(x[i]) - 39 * f(x[i+1]) + 12 * f(x[i+2]) - f(x[i+3]) ) / (6 * h^4)

Or

f4thderiv7ptcentrale1(f, x, i, h) = ( -1/6 * f(x[i-3]) + 2 * f(x[i-2]) - 13/2 * f(x[i-1]) + 28/3 * f(x[i]) - 13/2 * f(x[i+1]) + 2 * f(x[i+2]) - 1/6 * f(x[i+3]) ) / h^4

Or

f4thderiv7ptcentrald(f, x, i, h) = ( -0.166667 * f(x[i-3]) + 2 * f(x[i-2]) - 6.500000 * f(x[i-1]) + 9.333333 * f(x[i]) - 6.500000 * f(x[i+1]) + 2 * f(x[i+2]) - 0.166667 * f(x[i+3]) ) / h^4

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

7 * f(x[i-4]) - 96 * f(x[i-3]) + 676 * f(x[i-2]) - 1952 * f(x[i-1]) + 2730 * f(x[i]) - 1952 * f(x[i+1]) + 676 * f(x[i+2]) - 96 * f(x[i+3]) + 7 * f(x[i+4]) =
240 * f^(4)(x[i]) * h^4 + 82/63 * f^(10)(x[i]) * h^10 + 13/45 * f^(12)(x[i]) * h^12 + 13/396 * f^(14)(x[i]) * h^14 + 3203/1289925 * f^(16)(x[i]) * h^16 + ...

The exact formula:

f^(4)(x[i]) = ( 7 * f(x[i-4]) - 96 * f(x[i-3]) + 676 * f(x[i-2]) - 1952 * f(x[i-1]) + 2730 * f(x[i]) - 1952 * f(x[i+1]) + 676 * f(x[i+2]) - 96 * f(x[i+3]) + 7 * f(x[i+4]) ) / (240 * h^4) + O(h^6)

Or

f^(4)(x[i]) = ( 7/240 * f(x[i-4]) - 2/5 * f(x[i-3]) + 169/60 * f(x[i-2]) - 122/15 * f(x[i-1]) + 91/8 * f(x[i]) - 122/15 * f(x[i+1]) + 169/60 * f(x[i+2]) - 2/5 * f(x[i+3]) + 7/240 * f(x[i+4]) ) / h^4 + O(h^6)

Julia function:

f4thderiv9ptcentrale(f, x, i, h) = ( 7 * f(x[i-4]) - 96 * f(x[i-3]) + 676 * f(x[i-2]) - 1952 * f(x[i-1]) + 2730 * f(x[i]) - 1952 * f(x[i+1]) + 676 * f(x[i+2]) - 96 * f(x[i+3]) + 7 * f(x[i+4]) ) / (240 * h^4)

Or

f4thderiv9ptcentrale1(f, x, i, h) = ( 7/240 * f(x[i-4]) - 2/5 * f(x[i-3]) + 169/60 * f(x[i-2]) - 122/15 * f(x[i-1]) + 91/8 * f(x[i]) - 122/15 * f(x[i+1]) + 169/60 * f(x[i+2]) - 2/5 * f(x[i+3]) + 7/240 * f(x[i+4]) ) / h^4

Or

f4thderiv9ptcentrald(f, x, i, h) = ( 0.029167 * f(x[i-4]) - 0.400000 * f(x[i-3]) + 2.816667 * f(x[i-2]) - 8.133333 * f(x[i-1]) + 11.375000 * f(x[i]) - 8.133333 * f(x[i+1]) + 2.816667 * f(x[i+2]) - 0.400000 * f(x[i+3]) + 0.029167 * f(x[i+4]) ) / h^4

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 5 * f(x[i+1]) - 4 * f(x[i+2]) + f(x[i+3]) =
2 * f^(5)(x[i]) * h^5 + 2/3 * f^(7)(x[i]) * h^7 + 7/72 * f^(9)(x[i]) * h^9 + 8/945 * f^(11)(x[i]) * h^11 + 13/25920 * f^(13)(x[i]) * h^13 + ...

The exact formula:

f^(5)(x[i]) = ( -f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 5 * f(x[i+1]) - 4 * f(x[i+2]) + f(x[i+3]) ) / (2 * h^5) + O(h^2)

Or

f^(5)(x[i]) = ( -1/2 * f(x[i-3]) + 2 * f(x[i-2]) - 5/2 * f(x[i-1]) + 5/2 * f(x[i+1]) - 2 * f(x[i+2]) + 1/2 * f(x[i+3]) ) / h^5 + O(h^2)

Julia function:

f5thderiv6ptcentrale(f, x, i, h) = ( -f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 5 * f(x[i+1]) - 4 * f(x[i+2]) + f(x[i+3]) ) / (2 * h^5)

Or

f5thderiv6ptcentrale1(f, x, i, h) = ( -1/2 * f(x[i-3]) + 2 * f(x[i-2]) - 5/2 * f(x[i-1]) + 5/2 * f(x[i+1]) - 2 * f(x[i+2]) + 1/2 * f(x[i+3]) ) / h^5

Or

f5thderiv6ptcentrald(f, x, i, h) = ( -0.500000 * f(x[i-3]) + 2 * f(x[i-2]) - 2.500000 * f(x[i-1]) + 2.500000 * f(x[i+1]) - 2 * f(x[i+2]) + 0.500000 * f(x[i+3]) ) / h^5

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-4]) - 9 * f(x[i-3]) + 26 * f(x[i-2]) - 29 * f(x[i-1]) + 29 * f(x[i+1]) - 26 * f(x[i+2]) + 9 * f(x[i+3]) - f(x[i+4]) =
6 * f^(5)(x[i]) * h^5 - 13/24 * f^(9)(x[i]) * h^9 - 67/504 * f^(11)(x[i]) * h^11 - 49/2880 * f^(13)(x[i]) * h^13 - 481/332640 * f^(15)(x[i]) * h^15 + ...

The exact formula:

f^(5)(x[i]) = ( f(x[i-4]) - 9 * f(x[i-3]) + 26 * f(x[i-2]) - 29 * f(x[i-1]) + 29 * f(x[i+1]) - 26 * f(x[i+2]) + 9 * f(x[i+3]) - f(x[i+4]) ) / (6 * h^5) + O(h^4)

Or

f^(5)(x[i]) = ( 1/6 * f(x[i-4]) - 3/2 * f(x[i-3]) + 13/3 * f(x[i-2]) - 29/6 * f(x[i-1]) + 29/6 * f(x[i+1]) - 13/3 * f(x[i+2]) + 3/2 * f(x[i+3]) - 1/6 * f(x[i+4]) ) / h^5 + O(h^4)

Julia function:

f5thderiv8ptcentrale(f, x, i, h) = ( f(x[i-4]) - 9 * f(x[i-3]) + 26 * f(x[i-2]) - 29 * f(x[i-1]) + 29 * f(x[i+1]) - 26 * f(x[i+2]) + 9 * f(x[i+3]) - f(x[i+4]) ) / (6 * h^5)

Or

f5thderiv8ptcentrale1(f, x, i, h) = ( 1/6 * f(x[i-4]) - 3/2 * f(x[i-3]) + 13/3 * f(x[i-2]) - 29/6 * f(x[i-1]) + 29/6 * f(x[i+1]) - 13/3 * f(x[i+2]) + 3/2 * f(x[i+3]) - 1/6 * f(x[i+4]) ) / h^5

Or

f5thderiv8ptcentrald(f, x, i, h) = ( 0.166667 * f(x[i-4]) - 1.500000 * f(x[i-3]) + 4.333333 * f(x[i-2]) - 4.833333 * f(x[i-1]) + 4.833333 * f(x[i+1]) - 4.333333 * f(x[i+2]) + 1.500000 * f(x[i+3]) - 0.166667 * f(x[i+4]) ) / h^5

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-13 * f(x[i-5]) + 152 * f(x[i-4]) - 783 * f(x[i-3]) + 1872 * f(x[i-2]) - 1938 * f(x[i-1]) + 1938 * f(x[i+1]) - 1872 * f(x[i+2]) + 783 * f(x[i+3]) - 152 * f(x[i+4]) + 13 * f(x[i+5]) =
288 * f^(5)(x[i]) * h^5 + 139/21 * f^(11)(x[i]) * h^11 + 133/60 * f^(13)(x[i]) * h^13 + 247/660 * f^(15)(x[i]) * h^15 + 1146227/27518400 * f^(17)(x[i]) * h^17 + ...

The exact formula:

f^(5)(x[i]) = ( -13 * f(x[i-5]) + 152 * f(x[i-4]) - 783 * f(x[i-3]) + 1872 * f(x[i-2]) - 1938 * f(x[i-1]) + 1938 * f(x[i+1]) - 1872 * f(x[i+2]) + 783 * f(x[i+3]) - 152 * f(x[i+4]) + 13 * f(x[i+5]) ) / (288 * h^5) + O(h^6)

Or

f^(5)(x[i]) = ( -13/288 * f(x[i-5]) + 19/36 * f(x[i-4]) - 87/32 * f(x[i-3]) + 13/2 * f(x[i-2]) - 323/48 * f(x[i-1]) + 323/48 * f(x[i+1]) - 13/2 * f(x[i+2]) + 87/32 * f(x[i+3]) - 19/36 * f(x[i+4]) + 13/288 * f(x[i+5]) ) / h^5 + O(h^6)

Julia function:

f5thderiv10ptcentrale(f, x, i, h) = ( -13 * f(x[i-5]) + 152 * f(x[i-4]) - 783 * f(x[i-3]) + 1872 * f(x[i-2]) - 1938 * f(x[i-1]) + 1938 * f(x[i+1]) - 1872 * f(x[i+2]) + 783 * f(x[i+3]) - 152 * f(x[i+4]) + 13 * f(x[i+5]) ) / (288 * h^5)

Or

f5thderiv10ptcentrale1(f, x, i, h) = ( -13/288 * f(x[i-5]) + 19/36 * f(x[i-4]) - 87/32 * f(x[i-3]) + 13/2 * f(x[i-2]) - 323/48 * f(x[i-1]) + 323/48 * f(x[i+1]) - 13/2 * f(x[i+2]) + 87/32 * f(x[i+3]) - 19/36 * f(x[i+4]) + 13/288 * f(x[i+5]) ) / h^5

Or

f5thderiv10ptcentrald(f, x, i, h) = ( -0.045139 * f(x[i-5]) + 0.527778 * f(x[i-4]) - 2.718750 * f(x[i-3]) + 6.500000 * f(x[i-2]) - 6.729167 * f(x[i-1]) + 6.729167 * f(x[i+1]) - 6.500000 * f(x[i+2]) + 2.718750 * f(x[i+3]) - 0.527778 * f(x[i+4]) + 0.045139 * f(x[i+5]) ) / h^5

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

f(x[i-3]) - 6 * f(x[i-2]) + 15 * f(x[i-1]) - 20 * f(x[i]) + 15 * f(x[i+1]) - 6 * f(x[i+2]) + f(x[i+3]) =
f^(6)(x[i]) * h^6 + 1/4 * f^(8)(x[i]) * h^8 + 7/240 * f^(10)(x[i]) * h^10 + 2/945 * f^(12)(x[i]) * h^12 + 13/120960 * f^(14)(x[i]) * h^14 + ...

The exact formula:

f^(6)(x[i]) = ( f(x[i-3]) - 6 * f(x[i-2]) + 15 * f(x[i-1]) - 20 * f(x[i]) + 15 * f(x[i+1]) - 6 * f(x[i+2]) + f(x[i+3]) ) / h^6 + O(h^2)

Julia function:

f6thderiv7ptcentral(f, x, i, h) = ( f(x[i-3]) - 6 * f(x[i-2]) + 15 * f(x[i-1]) - 20 * f(x[i]) + 15 * f(x[i+1]) - 6 * f(x[i+2]) + f(x[i+3]) ) / h^6

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

-f(x[i-4]) + 12 * f(x[i-3]) - 52 * f(x[i-2]) + 116 * f(x[i-1]) - 150 * f(x[i]) + 116 * f(x[i+1]) - 52 * f(x[i+2]) + 12 * f(x[i+3]) - f(x[i+4]) =
4 * f^(6)(x[i]) * h^6 - 13/60 * f^(10)(x[i]) * h^10 - 67/1512 * f^(12)(x[i]) * h^12 - 7/1440 * f^(14)(x[i]) * h^14 - 481/1330560 * f^(16)(x[i]) * h^16 + ...

The exact formula:

f^(6)(x[i]) = ( -f(x[i-4]) + 12 * f(x[i-3]) - 52 * f(x[i-2]) + 116 * f(x[i-1]) - 150 * f(x[i]) + 116 * f(x[i+1]) - 52 * f(x[i+2]) + 12 * f(x[i+3]) - f(x[i+4]) ) / (4 * h^6) + O(h^4)

Or

f^(6)(x[i]) = ( -1/4 * f(x[i-4]) + 3 * f(x[i-3]) - 13 * f(x[i-2]) + 29 * f(x[i-1]) - 75/2 * f(x[i]) + 29 * f(x[i+1]) - 13 * f(x[i+2]) + 3 * f(x[i+3]) - 1/4 * f(x[i+4]) ) / h^6 + O(h^4)

Julia function:

f6thderiv9ptcentrale(f, x, i, h) = ( -f(x[i-4]) + 12 * f(x[i-3]) - 52 * f(x[i-2]) + 116 * f(x[i-1]) - 150 * f(x[i]) + 116 * f(x[i+1]) - 52 * f(x[i+2]) + 12 * f(x[i+3]) - f(x[i+4]) ) / (4 * h^6)

Or

f6thderiv9ptcentrale1(f, x, i, h) = ( -1/4 * f(x[i-4]) + 3 * f(x[i-3]) - 13 * f(x[i-2]) + 29 * f(x[i-1]) - 75/2 * f(x[i]) + 29 * f(x[i+1]) - 13 * f(x[i+2]) + 3 * f(x[i+3]) - 1/4 * f(x[i+4]) ) / h^6

Or

f6thderiv9ptcentrald(f, x, i, h) = ( -0.250000 * f(x[i-4]) + 3 * f(x[i-3]) - 13 * f(x[i-2]) + 29 * f(x[i-1]) - 37.500000 * f(x[i]) + 29 * f(x[i+1]) - 13 * f(x[i+2]) + 3 * f(x[i+3]) - 0.250000 * f(x[i+4]) ) / h^6

The following formula passed all tests: sum of coefs being zero, symmetry of coefs about x[i], etc.

Computing result:

13 * f(x[i-5]) - 190 * f(x[i-4]) + 1305 * f(x[i-3]) - 4680 * f(x[i-2]) + 9690 * f(x[i-1]) - 12276 * f(x[i]) + 9690 * f(x[i+1]) - 4680 * f(x[i+2]) + 1305 * f(x[i+3]) - 190 * f(x[i+4]) + 13 * f(x[i+5]) =
240 * f^(6)(x[i]) * h^6 + 695/252 * f^(12)(x[i]) * h^12 + 19/24 * f^(14)(x[i]) * h^14 + 247/2112 * f^(16)(x[i]) * h^16 + 1146227/99066240 * f^(18)(x[i]) * h^18 + ...

The exact formula:

f^(6)(x[i]) = ( 13 * f(x[i-5]) - 190 * f(x[i-4]) + 1305 * f(x[i-3]) - 4680 * f(x[i-2]) + 9690 * f(x[i-1]) - 12276 * f(x[i]) + 9690 * f(x[i+1]) - 4680 * f(x[i+2]) + 1305 * f(x[i+3]) - 190 * f(x[i+4]) + 13 * f(x[i+5]) ) / (240 * h^6) + O(h^6)

Or

f^(6)(x[i]) = ( 13/240 * f(x[i-5]) - 19/24 * f(x[i-4]) + 87/16 * f(x[i-3]) - 39/2 * f(x[i-2]) + 323/8 * f(x[i-1]) - 1023/20 * f(x[i]) + 323/8 * f(x[i+1]) - 39/2 * f(x[i+2]) + 87/16 * f(x[i+3]) - 19/24 * f(x[i+4]) + 13/240 * f(x[i+5]) ) / h^6 + O(h^6)

Julia function:

f6thderiv11ptcentrale(f, x, i, h) = ( 13 * f(x[i-5]) - 190 * f(x[i-4]) + 1305 * f(x[i-3]) - 4680 * f(x[i-2]) + 9690 * f(x[i-1]) - 12276 * f(x[i]) + 9690 * f(x[i+1]) - 4680 * f(x[i+2]) + 1305 * f(x[i+3]) - 190 * f(x[i+4]) + 13 * f(x[i+5]) ) / (240 * h^6)

Or

f6thderiv11ptcentrale1(f, x, i, h) = ( 13/240 * f(x[i-5]) - 19/24 * f(x[i-4]) + 87/16 * f(x[i-3]) - 39/2 * f(x[i-2]) + 323/8 * f(x[i-1]) - 1023/20 * f(x[i]) + 323/8 * f(x[i+1]) - 39/2 * f(x[i+2]) + 87/16 * f(x[i+3]) - 19/24 * f(x[i+4]) + 13/240 * f(x[i+5]) ) / h^6

Or

f6thderiv11ptcentrald(f, x, i, h) = ( 0.054167 * f(x[i-5]) - 0.791667 * f(x[i-4]) + 5.437500 * f(x[i-3]) - 19.500000 * f(x[i-2]) + 40.375000 * f(x[i-1]) - 51.150000 * f(x[i]) + 40.375000 * f(x[i+1]) - 19.500000 * f(x[i+2]) + 5.437500 * f(x[i+3]) - 0.791667 * f(x[i+4]) + 0.054167 * f(x[i+5]) ) / h^6

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-f(x[i]) + f(x[i+1]) =
f'(x[i]) * h + 1/2 * f''(x[i]) * h^2 + 1/6 * f'''(x[i]) * h^3 + 1/24 * f^(4)(x[i]) * h^4 + 1/120 * f^(5)(x[i]) * h^5 + ...

The exact formula:

f'(x[i]) = ( -f(x[i]) + f(x[i+1]) ) / h + O(h)

Julia function:

f1stderiv2ptforward(f, x, i, h) = ( -f(x[i]) + f(x[i+1]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-3 * f(x[i]) + 4 * f(x[i+1]) - f(x[i+2]) =
2 * f'(x[i]) * h - 2/3 * f'''(x[i]) * h^3 - 1/2 * f^(4)(x[i]) * h^4 - 7/30 * f^(5)(x[i]) * h^5 - 1/12 * f^(6)(x[i]) * h^6 + ...

The exact formula:

f'(x[i]) = ( -3 * f(x[i]) + 4 * f(x[i+1]) - f(x[i+2]) ) / (2 * h) + O(h^2)

Or

f'(x[i]) = ( -3/2 * f(x[i]) + 2 * f(x[i+1]) - 1/2 * f(x[i+2]) ) / h + O(h^2)

Julia function:

f1stderiv3ptforwarde(f, x, i, h) = ( -3 * f(x[i]) + 4 * f(x[i+1]) - f(x[i+2]) ) / (2 * h)

Or

f1stderiv3ptforwarde1(f, x, i, h) = ( -3/2 * f(x[i]) + 2 * f(x[i+1]) - 1/2 * f(x[i+2]) ) / h

Or

f1stderiv3ptforwardd(f, x, i, h) = ( -1.500000 * f(x[i]) + 2 * f(x[i+1]) - 0.500000 * f(x[i+2]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-11 * f(x[i]) + 18 * f(x[i+1]) - 9 * f(x[i+2]) + 2 * f(x[i+3]) =
6 * f'(x[i]) * h + 3/2 * f^(4)(x[i]) * h^4 + 9/5 * f^(5)(x[i]) * h^5 + 5/4 * f^(6)(x[i]) * h^6 + 9/14 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'(x[i]) = ( -11 * f(x[i]) + 18 * f(x[i+1]) - 9 * f(x[i+2]) + 2 * f(x[i+3]) ) / (6 * h) + O(h^3)

Or

f'(x[i]) = ( -11/6 * f(x[i]) + 3 * f(x[i+1]) - 3/2 * f(x[i+2]) + 1/3 * f(x[i+3]) ) / h + O(h^3)

Julia function:

f1stderiv4ptforwarde(f, x, i, h) = ( -11 * f(x[i]) + 18 * f(x[i+1]) - 9 * f(x[i+2]) + 2 * f(x[i+3]) ) / (6 * h)

Or

f1stderiv4ptforwarde1(f, x, i, h) = ( -11/6 * f(x[i]) + 3 * f(x[i+1]) - 3/2 * f(x[i+2]) + 1/3 * f(x[i+3]) ) / h

Or

f1stderiv4ptforwardd(f, x, i, h) = ( -1.833333 * f(x[i]) + 3 * f(x[i+1]) - 1.500000 * f(x[i+2]) + 0.333333 * f(x[i+3]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-25 * f(x[i]) + 48 * f(x[i+1]) - 36 * f(x[i+2]) + 16 * f(x[i+3]) - 3 * f(x[i+4]) =
12 * f'(x[i]) * h - 12/5 * f^(5)(x[i]) * h^5 - 4 * f^(6)(x[i]) * h^6 - 26/7 * f^(7)(x[i]) * h^7 - 5/2 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f'(x[i]) = ( -25 * f(x[i]) + 48 * f(x[i+1]) - 36 * f(x[i+2]) + 16 * f(x[i+3]) - 3 * f(x[i+4]) ) / (12 * h) + O(h^4)

Or

f'(x[i]) = ( -25/12 * f(x[i]) + 4 * f(x[i+1]) - 3 * f(x[i+2]) + 4/3 * f(x[i+3]) - 1/4 * f(x[i+4]) ) / h + O(h^4)

Julia function:

f1stderiv5ptforwarde(f, x, i, h) = ( -25 * f(x[i]) + 48 * f(x[i+1]) - 36 * f(x[i+2]) + 16 * f(x[i+3]) - 3 * f(x[i+4]) ) / (12 * h)

Or

f1stderiv5ptforwarde1(f, x, i, h) = ( -25/12 * f(x[i]) + 4 * f(x[i+1]) - 3 * f(x[i+2]) + 4/3 * f(x[i+3]) - 1/4 * f(x[i+4]) ) / h

Or

f1stderiv5ptforwardd(f, x, i, h) = ( -2.083333 * f(x[i]) + 4 * f(x[i+1]) - 3 * f(x[i+2]) + 1.333333 * f(x[i+3]) - 0.250000 * f(x[i+4]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-137 * f(x[i]) + 300 * f(x[i+1]) - 300 * f(x[i+2]) + 200 * f(x[i+3]) - 75 * f(x[i+4]) + 12 * f(x[i+5]) =
60 * f'(x[i]) * h + 10 * f^(6)(x[i]) * h^6 + 150/7 * f^(7)(x[i]) * h^7 + 25 * f^(8)(x[i]) * h^8 + 125/6 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f'(x[i]) = ( -137 * f(x[i]) + 300 * f(x[i+1]) - 300 * f(x[i+2]) + 200 * f(x[i+3]) - 75 * f(x[i+4]) + 12 * f(x[i+5]) ) / (60 * h) + O(h^5)

Or

f'(x[i]) = ( -137/60 * f(x[i]) + 5 * f(x[i+1]) - 5 * f(x[i+2]) + 10/3 * f(x[i+3]) - 5/4 * f(x[i+4]) + 1/5 * f(x[i+5]) ) / h + O(h^5)

Julia function:

f1stderiv6ptforwarde(f, x, i, h) = ( -137 * f(x[i]) + 300 * f(x[i+1]) - 300 * f(x[i+2]) + 200 * f(x[i+3]) - 75 * f(x[i+4]) + 12 * f(x[i+5]) ) / (60 * h)

Or

f1stderiv6ptforwarde1(f, x, i, h) = ( -137/60 * f(x[i]) + 5 * f(x[i+1]) - 5 * f(x[i+2]) + 10/3 * f(x[i+3]) - 5/4 * f(x[i+4]) + 1/5 * f(x[i+5]) ) / h

Or

f1stderiv6ptforwardd(f, x, i, h) = ( -2.283333 * f(x[i]) + 5 * f(x[i+1]) - 5 * f(x[i+2]) + 3.333333 * f(x[i+3]) - 1.250000 * f(x[i+4]) + 0.200000 * f(x[i+5]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-147 * f(x[i]) + 360 * f(x[i+1]) - 450 * f(x[i+2]) + 400 * f(x[i+3]) - 225 * f(x[i+4]) + 72 * f(x[i+5]) - 10 * f(x[i+6]) =
60 * f'(x[i]) * h - 60/7 * f^(7)(x[i]) * h^7 - 45/2 * f^(8)(x[i]) * h^8 - 95/3 * f^(9)(x[i]) * h^9 - 63/2 * f^(10)(x[i]) * h^10 + ...

The exact formula:

f'(x[i]) = ( -147 * f(x[i]) + 360 * f(x[i+1]) - 450 * f(x[i+2]) + 400 * f(x[i+3]) - 225 * f(x[i+4]) + 72 * f(x[i+5]) - 10 * f(x[i+6]) ) / (60 * h) + O(h^6)

Or

f'(x[i]) = ( -49/20 * f(x[i]) + 6 * f(x[i+1]) - 15/2 * f(x[i+2]) + 20/3 * f(x[i+3]) - 15/4 * f(x[i+4]) + 6/5 * f(x[i+5]) - 1/6 * f(x[i+6]) ) / h + O(h^6)

Julia function:

f1stderiv7ptforwarde(f, x, i, h) = ( -147 * f(x[i]) + 360 * f(x[i+1]) - 450 * f(x[i+2]) + 400 * f(x[i+3]) - 225 * f(x[i+4]) + 72 * f(x[i+5]) - 10 * f(x[i+6]) ) / (60 * h)

Or

f1stderiv7ptforwarde1(f, x, i, h) = ( -49/20 * f(x[i]) + 6 * f(x[i+1]) - 15/2 * f(x[i+2]) + 20/3 * f(x[i+3]) - 15/4 * f(x[i+4]) + 6/5 * f(x[i+5]) - 1/6 * f(x[i+6]) ) / h

Or

f1stderiv7ptforwardd(f, x, i, h) = ( -2.450000 * f(x[i]) + 6 * f(x[i+1]) - 7.500000 * f(x[i+2]) + 6.666667 * f(x[i+3]) - 3.750000 * f(x[i+4]) + 1.200000 * f(x[i+5]) - 0.166667 * f(x[i+6]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

f(x[i]) - 2 * f(x[i+1]) + f(x[i+2]) =
f''(x[i]) * h^2 + f'''(x[i]) * h^3 + 7/12 * f^(4)(x[i]) * h^4 + 1/4 * f^(5)(x[i]) * h^5 + 31/360 * f^(6)(x[i]) * h^6 + ...

The exact formula:

f''(x[i]) = ( f(x[i]) - 2 * f(x[i+1]) + f(x[i+2]) ) / h^2 + O(h)

Julia function:

f2ndderiv3ptforward(f, x, i, h) = ( f(x[i]) - 2 * f(x[i+1]) + f(x[i+2]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

2 * f(x[i]) - 5 * f(x[i+1]) + 4 * f(x[i+2]) - f(x[i+3]) =
f''(x[i]) * h^2 - 11/12 * f^(4)(x[i]) * h^4 - f^(5)(x[i]) * h^5 - 239/360 * f^(6)(x[i]) * h^6 - 1/3 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f''(x[i]) = ( 2 * f(x[i]) - 5 * f(x[i+1]) + 4 * f(x[i+2]) - f(x[i+3]) ) / h^2 + O(h^2)

Julia function:

f2ndderiv4ptforward(f, x, i, h) = ( 2 * f(x[i]) - 5 * f(x[i+1]) + 4 * f(x[i+2]) - f(x[i+3]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

35 * f(x[i]) - 104 * f(x[i+1]) + 114 * f(x[i+2]) - 56 * f(x[i+3]) + 11 * f(x[i+4]) =
12 * f''(x[i]) * h^2 + 10 * f^(5)(x[i]) * h^5 + 238/15 * f^(6)(x[i]) * h^6 + 43/3 * f^(7)(x[i]) * h^7 + 797/84 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f''(x[i]) = ( 35 * f(x[i]) - 104 * f(x[i+1]) + 114 * f(x[i+2]) - 56 * f(x[i+3]) + 11 * f(x[i+4]) ) / (12 * h^2) + O(h^3)

Or

f''(x[i]) = ( 35/12 * f(x[i]) - 26/3 * f(x[i+1]) + 19/2 * f(x[i+2]) - 14/3 * f(x[i+3]) + 11/12 * f(x[i+4]) ) / h^2 + O(h^3)

Julia function:

f2ndderiv5ptforwarde(f, x, i, h) = ( 35 * f(x[i]) - 104 * f(x[i+1]) + 114 * f(x[i+2]) - 56 * f(x[i+3]) + 11 * f(x[i+4]) ) / (12 * h^2)

Or

f2ndderiv5ptforwarde1(f, x, i, h) = ( 35/12 * f(x[i]) - 26/3 * f(x[i+1]) + 19/2 * f(x[i+2]) - 14/3 * f(x[i+3]) + 11/12 * f(x[i+4]) ) / h^2

Or

f2ndderiv5ptforwardd(f, x, i, h) = ( 2.916667 * f(x[i]) - 8.666667 * f(x[i+1]) + 9.500000 * f(x[i+2]) - 4.666667 * f(x[i+3]) + 0.916667 * f(x[i+4]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

45 * f(x[i]) - 154 * f(x[i+1]) + 214 * f(x[i+2]) - 156 * f(x[i+3]) + 61 * f(x[i+4]) - 10 * f(x[i+5]) =
12 * f''(x[i]) * h^2 - 137/15 * f^(6)(x[i]) * h^6 - 19 * f^(7)(x[i]) * h^7 - 457/21 * f^(8)(x[i]) * h^8 - 215/12 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f''(x[i]) = ( 45 * f(x[i]) - 154 * f(x[i+1]) + 214 * f(x[i+2]) - 156 * f(x[i+3]) + 61 * f(x[i+4]) - 10 * f(x[i+5]) ) / (12 * h^2) + O(h^4)

Or

f''(x[i]) = ( 15/4 * f(x[i]) - 77/6 * f(x[i+1]) + 107/6 * f(x[i+2]) - 13 * f(x[i+3]) + 61/12 * f(x[i+4]) - 5/6 * f(x[i+5]) ) / h^2 + O(h^4)

Julia function:

f2ndderiv6ptforwarde(f, x, i, h) = ( 45 * f(x[i]) - 154 * f(x[i+1]) + 214 * f(x[i+2]) - 156 * f(x[i+3]) + 61 * f(x[i+4]) - 10 * f(x[i+5]) ) / (12 * h^2)

Or

f2ndderiv6ptforwarde1(f, x, i, h) = ( 15/4 * f(x[i]) - 77/6 * f(x[i+1]) + 107/6 * f(x[i+2]) - 13 * f(x[i+3]) + 61/12 * f(x[i+4]) - 5/6 * f(x[i+5]) ) / h^2

Or

f2ndderiv6ptforwardd(f, x, i, h) = ( 3.750000 * f(x[i]) - 12.833333 * f(x[i+1]) + 17.833333 * f(x[i+2]) - 13 * f(x[i+3]) + 5.083333 * f(x[i+4]) - 0.833333 * f(x[i+5]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

812 * f(x[i]) - 3132 * f(x[i+1]) + 5265 * f(x[i+2]) - 5080 * f(x[i+3]) + 2970 * f(x[i+4]) - 972 * f(x[i+5]) + 137 * f(x[i+6]) =
180 * f''(x[i]) * h^2 + 126 * f^(7)(x[i]) * h^7 + 9081/28 * f^(8)(x[i]) * h^8 + 901/2 * f^(9)(x[i]) * h^9 + 8881/20 * f^(10)(x[i]) * h^10 + ...

The exact formula:

f''(x[i]) = ( 812 * f(x[i]) - 3132 * f(x[i+1]) + 5265 * f(x[i+2]) - 5080 * f(x[i+3]) + 2970 * f(x[i+4]) - 972 * f(x[i+5]) + 137 * f(x[i+6]) ) / (180 * h^2) + O(h^5)

Or

f''(x[i]) = ( 203/45 * f(x[i]) - 87/5 * f(x[i+1]) + 117/4 * f(x[i+2]) - 254/9 * f(x[i+3]) + 33/2 * f(x[i+4]) - 27/5 * f(x[i+5]) + 137/180 * f(x[i+6]) ) / h^2 + O(h^5)

Julia function:

f2ndderiv7ptforwarde(f, x, i, h) = ( 812 * f(x[i]) - 3132 * f(x[i+1]) + 5265 * f(x[i+2]) - 5080 * f(x[i+3]) + 2970 * f(x[i+4]) - 972 * f(x[i+5]) + 137 * f(x[i+6]) ) / (180 * h^2)

Or

f2ndderiv7ptforwarde1(f, x, i, h) = ( 203/45 * f(x[i]) - 87/5 * f(x[i+1]) + 117/4 * f(x[i+2]) - 254/9 * f(x[i+3]) + 33/2 * f(x[i+4]) - 27/5 * f(x[i+5]) + 137/180 * f(x[i+6]) ) / h^2

Or

f2ndderiv7ptforwardd(f, x, i, h) = ( 4.511111 * f(x[i]) - 17.400000 * f(x[i+1]) + 29.250000 * f(x[i+2]) - 28.222222 * f(x[i+3]) + 16.500000 * f(x[i+4]) - 5.400000 * f(x[i+5]) + 0.761111 * f(x[i+6]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

938 * f(x[i]) - 4014 * f(x[i+1]) + 7911 * f(x[i+2]) - 9490 * f(x[i+3]) + 7380 * f(x[i+4]) - 3618 * f(x[i+5]) + 1019 * f(x[i+6]) - 126 * f(x[i+7]) =
180 * f''(x[i]) * h^2 - 3267/28 * f^(8)(x[i]) * h^8 - 358 * f^(9)(x[i]) * h^9 - 11699/20 * f^(10)(x[i]) * h^10 - 672 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f''(x[i]) = ( 938 * f(x[i]) - 4014 * f(x[i+1]) + 7911 * f(x[i+2]) - 9490 * f(x[i+3]) + 7380 * f(x[i+4]) - 3618 * f(x[i+5]) + 1019 * f(x[i+6]) - 126 * f(x[i+7]) ) / (180 * h^2) + O(h^6)

Or

f''(x[i]) = ( 469/90 * f(x[i]) - 223/10 * f(x[i+1]) + 879/20 * f(x[i+2]) - 949/18 * f(x[i+3]) + 41 * f(x[i+4]) - 201/10 * f(x[i+5]) + 1019/180 * f(x[i+6]) - 7/10 * f(x[i+7]) ) / h^2 + O(h^6)

Julia function:

f2ndderiv8ptforwarde(f, x, i, h) = ( 938 * f(x[i]) - 4014 * f(x[i+1]) + 7911 * f(x[i+2]) - 9490 * f(x[i+3]) + 7380 * f(x[i+4]) - 3618 * f(x[i+5]) + 1019 * f(x[i+6]) - 126 * f(x[i+7]) ) / (180 * h^2)

Or

f2ndderiv8ptforwarde1(f, x, i, h) = ( 469/90 * f(x[i]) - 223/10 * f(x[i+1]) + 879/20 * f(x[i+2]) - 949/18 * f(x[i+3]) + 41 * f(x[i+4]) - 201/10 * f(x[i+5]) + 1019/180 * f(x[i+6]) - 7/10 * f(x[i+7]) ) / h^2

Or

f2ndderiv8ptforwardd(f, x, i, h) = ( 5.211111 * f(x[i]) - 22.300000 * f(x[i+1]) + 43.950000 * f(x[i+2]) - 52.722222 * f(x[i+3]) + 41 * f(x[i+4]) - 20.100000 * f(x[i+5]) + 5.661111 * f(x[i+6]) - 0.700000 * f(x[i+7]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-f(x[i]) + 3 * f(x[i+1]) - 3 * f(x[i+2]) + f(x[i+3]) =
f'''(x[i]) * h^3 + 3/2 * f^(4)(x[i]) * h^4 + 5/4 * f^(5)(x[i]) * h^5 + 3/4 * f^(6)(x[i]) * h^6 + 43/120 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'''(x[i]) = ( -f(x[i]) + 3 * f(x[i+1]) - 3 * f(x[i+2]) + f(x[i+3]) ) / h^3 + O(h)

Julia function:

f3rdderiv4ptforward(f, x, i, h) = ( -f(x[i]) + 3 * f(x[i+1]) - 3 * f(x[i+2]) + f(x[i+3]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-5 * f(x[i]) + 18 * f(x[i+1]) - 24 * f(x[i+2]) + 14 * f(x[i+3]) - 3 * f(x[i+4]) =
2 * f'''(x[i]) * h^3 - 7/2 * f^(5)(x[i]) * h^5 - 5 * f^(6)(x[i]) * h^6 - 257/60 * f^(7)(x[i]) * h^7 - 11/4 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f'''(x[i]) = ( -5 * f(x[i]) + 18 * f(x[i+1]) - 24 * f(x[i+2]) + 14 * f(x[i+3]) - 3 * f(x[i+4]) ) / (2 * h^3) + O(h^2)

Or

f'''(x[i]) = ( -5/2 * f(x[i]) + 9 * f(x[i+1]) - 12 * f(x[i+2]) + 7 * f(x[i+3]) - 3/2 * f(x[i+4]) ) / h^3 + O(h^2)

Julia function:

f3rdderiv5ptforwarde(f, x, i, h) = ( -5 * f(x[i]) + 18 * f(x[i+1]) - 24 * f(x[i+2]) + 14 * f(x[i+3]) - 3 * f(x[i+4]) ) / (2 * h^3)

Or

f3rdderiv5ptforwarde1(f, x, i, h) = ( -5/2 * f(x[i]) + 9 * f(x[i+1]) - 12 * f(x[i+2]) + 7 * f(x[i+3]) - 3/2 * f(x[i+4]) ) / h^3

Or

f3rdderiv5ptforwardd(f, x, i, h) = ( -2.500000 * f(x[i]) + 9 * f(x[i+1]) - 12 * f(x[i+2]) + 7 * f(x[i+3]) - 1.500000 * f(x[i+4]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-17 * f(x[i]) + 71 * f(x[i+1]) - 118 * f(x[i+2]) + 98 * f(x[i+3]) - 41 * f(x[i+4]) + 7 * f(x[i+5]) =
4 * f'''(x[i]) * h^3 + 15/2 * f^(6)(x[i]) * h^6 + 443/30 * f^(7)(x[i]) * h^7 + 131/8 * f^(8)(x[i]) * h^8 + 19969/1512 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f'''(x[i]) = ( -17 * f(x[i]) + 71 * f(x[i+1]) - 118 * f(x[i+2]) + 98 * f(x[i+3]) - 41 * f(x[i+4]) + 7 * f(x[i+5]) ) / (4 * h^3) + O(h^3)

Or

f'''(x[i]) = ( -17/4 * f(x[i]) + 71/4 * f(x[i+1]) - 59/2 * f(x[i+2]) + 49/2 * f(x[i+3]) - 41/4 * f(x[i+4]) + 7/4 * f(x[i+5]) ) / h^3 + O(h^3)

Julia function:

f3rdderiv6ptforwarde(f, x, i, h) = ( -17 * f(x[i]) + 71 * f(x[i+1]) - 118 * f(x[i+2]) + 98 * f(x[i+3]) - 41 * f(x[i+4]) + 7 * f(x[i+5]) ) / (4 * h^3)

Or

f3rdderiv6ptforwarde1(f, x, i, h) = ( -17/4 * f(x[i]) + 71/4 * f(x[i+1]) - 59/2 * f(x[i+2]) + 49/2 * f(x[i+3]) - 41/4 * f(x[i+4]) + 7/4 * f(x[i+5]) ) / h^3

Or

f3rdderiv6ptforwardd(f, x, i, h) = ( -4.250000 * f(x[i]) + 17.750000 * f(x[i+1]) - 29.500000 * f(x[i+2]) + 24.500000 * f(x[i+3]) - 10.250000 * f(x[i+4]) + 1.750000 * f(x[i+5]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-49 * f(x[i]) + 232 * f(x[i+1]) - 461 * f(x[i+2]) + 496 * f(x[i+3]) - 307 * f(x[i+4]) + 104 * f(x[i+5]) - 15 * f(x[i+6]) =
8 * f'''(x[i]) * h^3 - 232/15 * f^(7)(x[i]) * h^7 - 77/2 * f^(8)(x[i]) * h^8 - 19783/378 * f^(9)(x[i]) * h^9 - 305/6 * f^(10)(x[i]) * h^10 + ...

The exact formula:

f'''(x[i]) = ( -49 * f(x[i]) + 232 * f(x[i+1]) - 461 * f(x[i+2]) + 496 * f(x[i+3]) - 307 * f(x[i+4]) + 104 * f(x[i+5]) - 15 * f(x[i+6]) ) / (8 * h^3) + O(h^4)

Or

f'''(x[i]) = ( -49/8 * f(x[i]) + 29 * f(x[i+1]) - 461/8 * f(x[i+2]) + 62 * f(x[i+3]) - 307/8 * f(x[i+4]) + 13 * f(x[i+5]) - 15/8 * f(x[i+6]) ) / h^3 + O(h^4)

Julia function:

f3rdderiv7ptforwarde(f, x, i, h) = ( -49 * f(x[i]) + 232 * f(x[i+1]) - 461 * f(x[i+2]) + 496 * f(x[i+3]) - 307 * f(x[i+4]) + 104 * f(x[i+5]) - 15 * f(x[i+6]) ) / (8 * h^3)

Or

f3rdderiv7ptforwarde1(f, x, i, h) = ( -49/8 * f(x[i]) + 29 * f(x[i+1]) - 461/8 * f(x[i+2]) + 62 * f(x[i+3]) - 307/8 * f(x[i+4]) + 13 * f(x[i+5]) - 15/8 * f(x[i+6]) ) / h^3

Or

f3rdderiv7ptforwardd(f, x, i, h) = ( -6.125000 * f(x[i]) + 29 * f(x[i+1]) - 57.625000 * f(x[i+2]) + 62 * f(x[i+3]) - 38.375000 * f(x[i+4]) + 13 * f(x[i+5]) - 1.875000 * f(x[i+6]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-967 * f(x[i]) + 5104 * f(x[i+1]) - 11787 * f(x[i+2]) + 15560 * f(x[i+3]) - 12725 * f(x[i+4]) + 6432 * f(x[i+5]) - 1849 * f(x[i+6]) + 232 * f(x[i+7]) =
120 * f'''(x[i]) * h^3 + 469/2 * f^(8)(x[i]) * h^8 + 88657/126 * f^(9)(x[i]) * h^9 + 6793/6 * f^(10)(x[i]) * h^10 + 38593/30 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f'''(x[i]) = ( -967 * f(x[i]) + 5104 * f(x[i+1]) - 11787 * f(x[i+2]) + 15560 * f(x[i+3]) - 12725 * f(x[i+4]) + 6432 * f(x[i+5]) - 1849 * f(x[i+6]) + 232 * f(x[i+7]) ) / (120 * h^3) + O(h^5)

Or

f'''(x[i]) = ( -967/120 * f(x[i]) + 638/15 * f(x[i+1]) - 3929/40 * f(x[i+2]) + 389/3 * f(x[i+3]) - 2545/24 * f(x[i+4]) + 268/5 * f(x[i+5]) - 1849/120 * f(x[i+6]) + 29/15 * f(x[i+7]) ) / h^3 + O(h^5)

Julia function:

f3rdderiv8ptforwarde(f, x, i, h) = ( -967 * f(x[i]) + 5104 * f(x[i+1]) - 11787 * f(x[i+2]) + 15560 * f(x[i+3]) - 12725 * f(x[i+4]) + 6432 * f(x[i+5]) - 1849 * f(x[i+6]) + 232 * f(x[i+7]) ) / (120 * h^3)

Or

f3rdderiv8ptforwarde1(f, x, i, h) = ( -967/120 * f(x[i]) + 638/15 * f(x[i+1]) - 3929/40 * f(x[i+2]) + 389/3 * f(x[i+3]) - 2545/24 * f(x[i+4]) + 268/5 * f(x[i+5]) - 1849/120 * f(x[i+6]) + 29/15 * f(x[i+7]) ) / h^3

Or

f3rdderiv8ptforwardd(f, x, i, h) = ( -8.058333 * f(x[i]) + 42.533333 * f(x[i+1]) - 98.225000 * f(x[i+2]) + 129.666667 * f(x[i+3]) - 106.041667 * f(x[i+4]) + 53.600000 * f(x[i+5]) - 15.408333 * f(x[i+6]) + 1.933333 * f(x[i+7]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-2403 * f(x[i]) + 13960 * f(x[i+1]) - 36706 * f(x[i+2]) + 57384 * f(x[i+3]) - 58280 * f(x[i+4]) + 39128 * f(x[i+5]) - 16830 * f(x[i+6]) + 4216 * f(x[i+7]) - 469 * f(x[i+8]) =
240 * f'''(x[i]) * h^3 - 29531/63 * f^(9)(x[i]) * h^9 - 1644 * f^(10)(x[i]) * h^10 - 45827/15 * f^(11)(x[i]) * h^11 - 3976 * f^(12)(x[i]) * h^12 + ...

The exact formula:

f'''(x[i]) = ( -2403 * f(x[i]) + 13960 * f(x[i+1]) - 36706 * f(x[i+2]) + 57384 * f(x[i+3]) - 58280 * f(x[i+4]) + 39128 * f(x[i+5]) - 16830 * f(x[i+6]) + 4216 * f(x[i+7]) - 469 * f(x[i+8]) ) / (240 * h^3) + O(h^6)

Or

f'''(x[i]) = ( -801/80 * f(x[i]) + 349/6 * f(x[i+1]) - 18353/120 * f(x[i+2]) + 2391/10 * f(x[i+3]) - 1457/6 * f(x[i+4]) + 4891/30 * f(x[i+5]) - 561/8 * f(x[i+6]) + 527/30 * f(x[i+7]) - 469/240 * f(x[i+8]) ) / h^3 + O(h^6)

Julia function:

f3rdderiv9ptforwarde(f, x, i, h) = ( -2403 * f(x[i]) + 13960 * f(x[i+1]) - 36706 * f(x[i+2]) + 57384 * f(x[i+3]) - 58280 * f(x[i+4]) + 39128 * f(x[i+5]) - 16830 * f(x[i+6]) + 4216 * f(x[i+7]) - 469 * f(x[i+8]) ) / (240 * h^3)

Or

f3rdderiv9ptforwarde1(f, x, i, h) = ( -801/80 * f(x[i]) + 349/6 * f(x[i+1]) - 18353/120 * f(x[i+2]) + 2391/10 * f(x[i+3]) - 1457/6 * f(x[i+4]) + 4891/30 * f(x[i+5]) - 561/8 * f(x[i+6]) + 527/30 * f(x[i+7]) - 469/240 * f(x[i+8]) ) / h^3

Or

f3rdderiv9ptforwardd(f, x, i, h) = ( -10.012500 * f(x[i]) + 58.166667 * f(x[i+1]) - 152.941667 * f(x[i+2]) + 239.100000 * f(x[i+3]) - 242.833333 * f(x[i+4]) + 163.033333 * f(x[i+5]) - 70.125000 * f(x[i+6]) + 17.566667 * f(x[i+7]) - 1.954167 * f(x[i+8]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

f(x[i]) - 4 * f(x[i+1]) + 6 * f(x[i+2]) - 4 * f(x[i+3]) + f(x[i+4]) =
f^(4)(x[i]) * h^4 + 2 * f^(5)(x[i]) * h^5 + 13/6 * f^(6)(x[i]) * h^6 + 5/3 * f^(7)(x[i]) * h^7 + 81/80 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f^(4)(x[i]) = ( f(x[i]) - 4 * f(x[i+1]) + 6 * f(x[i+2]) - 4 * f(x[i+3]) + f(x[i+4]) ) / h^4 + O(h)

Julia function:

f4thderiv5ptforward(f, x, i, h) = ( f(x[i]) - 4 * f(x[i+1]) + 6 * f(x[i+2]) - 4 * f(x[i+3]) + f(x[i+4]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

3 * f(x[i]) - 14 * f(x[i+1]) + 26 * f(x[i+2]) - 24 * f(x[i+3]) + 11 * f(x[i+4]) - 2 * f(x[i+5]) =
f^(4)(x[i]) * h^4 - 17/6 * f^(6)(x[i]) * h^6 - 5 * f^(7)(x[i]) * h^7 - 419/80 * f^(8)(x[i]) * h^8 - 49/12 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f^(4)(x[i]) = ( 3 * f(x[i]) - 14 * f(x[i+1]) + 26 * f(x[i+2]) - 24 * f(x[i+3]) + 11 * f(x[i+4]) - 2 * f(x[i+5]) ) / h^4 + O(h^2)

Julia function:

f4thderiv6ptforward(f, x, i, h) = ( 3 * f(x[i]) - 14 * f(x[i+1]) + 26 * f(x[i+2]) - 24 * f(x[i+3]) + 11 * f(x[i+4]) - 2 * f(x[i+5]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

35 * f(x[i]) - 186 * f(x[i+1]) + 411 * f(x[i+2]) - 484 * f(x[i+3]) + 321 * f(x[i+4]) - 114 * f(x[i+5]) + 17 * f(x[i+6]) =
6 * f^(4)(x[i]) * h^4 + 21 * f^(7)(x[i]) * h^7 + 1973/40 * f^(8)(x[i]) * h^8 + 259/4 * f^(9)(x[i]) * h^9 + 30983/504 * f^(10)(x[i]) * h^10 + ...

The exact formula:

f^(4)(x[i]) = ( 35 * f(x[i]) - 186 * f(x[i+1]) + 411 * f(x[i+2]) - 484 * f(x[i+3]) + 321 * f(x[i+4]) - 114 * f(x[i+5]) + 17 * f(x[i+6]) ) / (6 * h^4) + O(h^3)

Or

f^(4)(x[i]) = ( 35/6 * f(x[i]) - 31 * f(x[i+1]) + 137/2 * f(x[i+2]) - 242/3 * f(x[i+3]) + 107/2 * f(x[i+4]) - 19 * f(x[i+5]) + 17/6 * f(x[i+6]) ) / h^4 + O(h^3)

Julia function:

f4thderiv7ptforwarde(f, x, i, h) = ( 35 * f(x[i]) - 186 * f(x[i+1]) + 411 * f(x[i+2]) - 484 * f(x[i+3]) + 321 * f(x[i+4]) - 114 * f(x[i+5]) + 17 * f(x[i+6]) ) / (6 * h^4)

Or

f4thderiv7ptforwarde1(f, x, i, h) = ( 35/6 * f(x[i]) - 31 * f(x[i+1]) + 137/2 * f(x[i+2]) - 242/3 * f(x[i+3]) + 107/2 * f(x[i+4]) - 19 * f(x[i+5]) + 17/6 * f(x[i+6]) ) / h^4

Or

f4thderiv7ptforwardd(f, x, i, h) = ( 5.833333 * f(x[i]) - 31 * f(x[i+1]) + 68.500000 * f(x[i+2]) - 80.666667 * f(x[i+3]) + 53.500000 * f(x[i+4]) - 19 * f(x[i+5]) + 2.833333 * f(x[i+6]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

56 * f(x[i]) - 333 * f(x[i+1]) + 852 * f(x[i+2]) - 1219 * f(x[i+3]) + 1056 * f(x[i+4]) - 555 * f(x[i+5]) + 164 * f(x[i+6]) - 21 * f(x[i+7]) =
6 * f^(4)(x[i]) * h^4 - 967/40 * f^(8)(x[i]) * h^8 - 70 * f^(9)(x[i]) * h^9 - 55453/504 * f^(10)(x[i]) * h^10 - 123 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f^(4)(x[i]) = ( 56 * f(x[i]) - 333 * f(x[i+1]) + 852 * f(x[i+2]) - 1219 * f(x[i+3]) + 1056 * f(x[i+4]) - 555 * f(x[i+5]) + 164 * f(x[i+6]) - 21 * f(x[i+7]) ) / (6 * h^4) + O(h^4)

Or

f^(4)(x[i]) = ( 28/3 * f(x[i]) - 111/2 * f(x[i+1]) + 142 * f(x[i+2]) - 1219/6 * f(x[i+3]) + 176 * f(x[i+4]) - 185/2 * f(x[i+5]) + 82/3 * f(x[i+6]) - 7/2 * f(x[i+7]) ) / h^4 + O(h^4)

Julia function:

f4thderiv8ptforwarde(f, x, i, h) = ( 56 * f(x[i]) - 333 * f(x[i+1]) + 852 * f(x[i+2]) - 1219 * f(x[i+3]) + 1056 * f(x[i+4]) - 555 * f(x[i+5]) + 164 * f(x[i+6]) - 21 * f(x[i+7]) ) / (6 * h^4)

Or

f4thderiv8ptforwarde1(f, x, i, h) = ( 28/3 * f(x[i]) - 111/2 * f(x[i+1]) + 142 * f(x[i+2]) - 1219/6 * f(x[i+3]) + 176 * f(x[i+4]) - 185/2 * f(x[i+5]) + 82/3 * f(x[i+6]) - 7/2 * f(x[i+7]) ) / h^4

Or

f4thderiv8ptforwardd(f, x, i, h) = ( 9.333333 * f(x[i]) - 55.500000 * f(x[i+1]) + 142 * f(x[i+2]) - 203.166667 * f(x[i+3]) + 176 * f(x[i+4]) - 92.500000 * f(x[i+5]) + 27.333333 * f(x[i+6]) - 3.500000 * f(x[i+7]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

3207 * f(x[i]) - 21056 * f(x[i+1]) + 61156 * f(x[i+2]) - 102912 * f(x[i+3]) + 109930 * f(x[i+4]) - 76352 * f(x[i+5]) + 33636 * f(x[i+6]) - 8576 * f(x[i+7]) + 967 * f(x[i+8]) =
240 * f^(4)(x[i]) * h^4 + 1068 * f^(9)(x[i]) * h^9 + 230410/63 * f^(10)(x[i]) * h^10 + 6684 * f^(11)(x[i]) * h^11 + 386713/45 * f^(12)(x[i]) * h^12 + ...

The exact formula:

f^(4)(x[i]) = ( 3207 * f(x[i]) - 21056 * f(x[i+1]) + 61156 * f(x[i+2]) - 102912 * f(x[i+3]) + 109930 * f(x[i+4]) - 76352 * f(x[i+5]) + 33636 * f(x[i+6]) - 8576 * f(x[i+7]) + 967 * f(x[i+8]) ) / (240 * h^4) + O(h^5)

Or

f^(4)(x[i]) = ( 1069/80 * f(x[i]) - 1316/15 * f(x[i+1]) + 15289/60 * f(x[i+2]) - 2144/5 * f(x[i+3]) + 10993/24 * f(x[i+4]) - 4772/15 * f(x[i+5]) + 2803/20 * f(x[i+6]) - 536/15 * f(x[i+7]) + 967/240 * f(x[i+8]) ) / h^4 + O(h^5)

Julia function:

f4thderiv9ptforwarde(f, x, i, h) = ( 3207 * f(x[i]) - 21056 * f(x[i+1]) + 61156 * f(x[i+2]) - 102912 * f(x[i+3]) + 109930 * f(x[i+4]) - 76352 * f(x[i+5]) + 33636 * f(x[i+6]) - 8576 * f(x[i+7]) + 967 * f(x[i+8]) ) / (240 * h^4)

Or

f4thderiv9ptforwarde1(f, x, i, h) = ( 1069/80 * f(x[i]) - 1316/15 * f(x[i+1]) + 15289/60 * f(x[i+2]) - 2144/5 * f(x[i+3]) + 10993/24 * f(x[i+4]) - 4772/15 * f(x[i+5]) + 2803/20 * f(x[i+6]) - 536/15 * f(x[i+7]) + 967/240 * f(x[i+8]) ) / h^4

Or

f4thderiv9ptforwardd(f, x, i, h) = ( 13.362500 * f(x[i]) - 87.733333 * f(x[i+1]) + 254.816667 * f(x[i+2]) - 428.800000 * f(x[i+3]) + 458.041667 * f(x[i+4]) - 318.133333 * f(x[i+5]) + 140.150000 * f(x[i+6]) - 35.733333 * f(x[i+7]) + 4.029167 * f(x[i+8]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-f(x[i-1]) + f(x[i]) =
f'(x[i]) * h - 1/2 * f''(x[i]) * h^2 + 1/6 * f'''(x[i]) * h^3 - 1/24 * f^(4)(x[i]) * h^4 + 1/120 * f^(5)(x[i]) * h^5 + ...

The exact formula:

f'(x[i]) = ( -f(x[i-1]) + f(x[i]) ) / h + O(h)

Julia function:

f1stderiv2ptbackward(f, x, i, h) = ( -f(x[i-1]) + f(x[i]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

f(x[i-2]) - 4 * f(x[i-1]) + 3 * f(x[i]) =
2 * f'(x[i]) * h - 2/3 * f'''(x[i]) * h^3 + 1/2 * f^(4)(x[i]) * h^4 - 7/30 * f^(5)(x[i]) * h^5 + 1/12 * f^(6)(x[i]) * h^6 + ...

The exact formula:

f'(x[i]) = ( f(x[i-2]) - 4 * f(x[i-1]) + 3 * f(x[i]) ) / (2 * h) + O(h^2)

Or

f'(x[i]) = ( 1/2 * f(x[i-2]) - 2 * f(x[i-1]) + 3/2 * f(x[i]) ) / h + O(h^2)

Julia function:

f1stderiv3ptbackwarde(f, x, i, h) = ( f(x[i-2]) - 4 * f(x[i-1]) + 3 * f(x[i]) ) / (2 * h)

Or

f1stderiv3ptbackwarde1(f, x, i, h) = ( 1/2 * f(x[i-2]) - 2 * f(x[i-1]) + 3/2 * f(x[i]) ) / h

Or

f1stderiv3ptbackwardd(f, x, i, h) = ( 0.500000 * f(x[i-2]) - 2 * f(x[i-1]) + 1.500000 * f(x[i]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-2 * f(x[i-3]) + 9 * f(x[i-2]) - 18 * f(x[i-1]) + 11 * f(x[i]) =
6 * f'(x[i]) * h - 3/2 * f^(4)(x[i]) * h^4 + 9/5 * f^(5)(x[i]) * h^5 - 5/4 * f^(6)(x[i]) * h^6 + 9/14 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'(x[i]) = ( -2 * f(x[i-3]) + 9 * f(x[i-2]) - 18 * f(x[i-1]) + 11 * f(x[i]) ) / (6 * h) + O(h^3)

Or

f'(x[i]) = ( -1/3 * f(x[i-3]) + 3/2 * f(x[i-2]) - 3 * f(x[i-1]) + 11/6 * f(x[i]) ) / h + O(h^3)

Julia function:

f1stderiv4ptbackwarde(f, x, i, h) = ( -2 * f(x[i-3]) + 9 * f(x[i-2]) - 18 * f(x[i-1]) + 11 * f(x[i]) ) / (6 * h)

Or

f1stderiv4ptbackwarde1(f, x, i, h) = ( -1/3 * f(x[i-3]) + 3/2 * f(x[i-2]) - 3 * f(x[i-1]) + 11/6 * f(x[i]) ) / h

Or

f1stderiv4ptbackwardd(f, x, i, h) = ( -0.333333 * f(x[i-3]) + 1.500000 * f(x[i-2]) - 3 * f(x[i-1]) + 1.833333 * f(x[i]) ) / h

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

f(x[i-2]) - 2 * f(x[i-1]) + f(x[i]) =
f''(x[i]) * h^2 - f'''(x[i]) * h^3 + 7/12 * f^(4)(x[i]) * h^4 - 1/4 * f^(5)(x[i]) * h^5 + 31/360 * f^(6)(x[i]) * h^6 + ...

The exact formula:

f''(x[i]) = ( f(x[i-2]) - 2 * f(x[i-1]) + f(x[i]) ) / h^2 + O(h)

Julia function:

f2ndderiv3ptbackward(f, x, i, h) = ( f(x[i-2]) - 2 * f(x[i-1]) + f(x[i]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 2 * f(x[i]) =
f''(x[i]) * h^2 - 11/12 * f^(4)(x[i]) * h^4 + f^(5)(x[i]) * h^5 - 239/360 * f^(6)(x[i]) * h^6 + 1/3 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f''(x[i]) = ( -f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 2 * f(x[i]) ) / h^2 + O(h^2)

Julia function:

f2ndderiv4ptbackward(f, x, i, h) = ( -f(x[i-3]) + 4 * f(x[i-2]) - 5 * f(x[i-1]) + 2 * f(x[i]) ) / h^2

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-f(x[i-3]) + 3 * f(x[i-2]) - 3 * f(x[i-1]) + f(x[i]) =
f'''(x[i]) * h^3 - 3/2 * f^(4)(x[i]) * h^4 + 5/4 * f^(5)(x[i]) * h^5 - 3/4 * f^(6)(x[i]) * h^6 + 43/120 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'''(x[i]) = ( -f(x[i-3]) + 3 * f(x[i-2]) - 3 * f(x[i-1]) + f(x[i]) ) / h^3 + O(h)

Julia function:

f3rdderiv4ptbackward(f, x, i, h) = ( -f(x[i-3]) + 3 * f(x[i-2]) - 3 * f(x[i-1]) + f(x[i]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

3 * f(x[i-4]) - 14 * f(x[i-3]) + 24 * f(x[i-2]) - 18 * f(x[i-1]) + 5 * f(x[i]) =
2 * f'''(x[i]) * h^3 - 7/2 * f^(5)(x[i]) * h^5 + 5 * f^(6)(x[i]) * h^6 - 257/60 * f^(7)(x[i]) * h^7 + 11/4 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f'''(x[i]) = ( 3 * f(x[i-4]) - 14 * f(x[i-3]) + 24 * f(x[i-2]) - 18 * f(x[i-1]) + 5 * f(x[i]) ) / (2 * h^3) + O(h^2)

Or

f'''(x[i]) = ( 3/2 * f(x[i-4]) - 7 * f(x[i-3]) + 12 * f(x[i-2]) - 9 * f(x[i-1]) + 5/2 * f(x[i]) ) / h^3 + O(h^2)

Julia function:

f3rdderiv5ptbackwarde(f, x, i, h) = ( 3 * f(x[i-4]) - 14 * f(x[i-3]) + 24 * f(x[i-2]) - 18 * f(x[i-1]) + 5 * f(x[i]) ) / (2 * h^3)

Or

f3rdderiv5ptbackwarde1(f, x, i, h) = ( 3/2 * f(x[i-4]) - 7 * f(x[i-3]) + 12 * f(x[i-2]) - 9 * f(x[i-1]) + 5/2 * f(x[i]) ) / h^3

Or

f3rdderiv5ptbackwardd(f, x, i, h) = ( 1.500000 * f(x[i-4]) - 7 * f(x[i-3]) + 12 * f(x[i-2]) - 9 * f(x[i-1]) + 2.500000 * f(x[i]) ) / h^3

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

f(x[i-4]) - 4 * f(x[i-3]) + 6 * f(x[i-2]) - 4 * f(x[i-1]) + f(x[i]) =
f^(4)(x[i]) * h^4 - 2 * f^(5)(x[i]) * h^5 + 13/6 * f^(6)(x[i]) * h^6 - 5/3 * f^(7)(x[i]) * h^7 + 81/80 * f^(8)(x[i]) * h^8 + ...

The exact formula:

f^(4)(x[i]) = ( f(x[i-4]) - 4 * f(x[i-3]) + 6 * f(x[i-2]) - 4 * f(x[i-1]) + f(x[i]) ) / h^4 + O(h)

Julia function:

f4thderiv5ptbackward(f, x, i, h) = ( f(x[i-4]) - 4 * f(x[i-3]) + 6 * f(x[i-2]) - 4 * f(x[i-1]) + f(x[i]) ) / h^4

The following formula passed all tests: sum of coefs being zero, etc.

Computing result:

-2 * f(x[i-5]) + 11 * f(x[i-4]) - 24 * f(x[i-3]) + 26 * f(x[i-2]) - 14 * f(x[i-1]) + 3 * f(x[i]) =
f^(4)(x[i]) * h^4 - 17/6 * f^(6)(x[i]) * h^6 + 5 * f^(7)(x[i]) * h^7 - 419/80 * f^(8)(x[i]) * h^8 + 49/12 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f^(4)(x[i]) = ( -2 * f(x[i-5]) + 11 * f(x[i-4]) - 24 * f(x[i-3]) + 26 * f(x[i-2]) - 14 * f(x[i-1]) + 3 * f(x[i]) ) / h^4 + O(h^2)

Julia function:

f4thderiv6ptbackward(f, x, i, h) = ( -2 * f(x[i-5]) + 11 * f(x[i-4]) - 24 * f(x[i-3]) + 26 * f(x[i-2]) - 14 * f(x[i-1]) + 3 * f(x[i]) ) / h^4

