You may start your work by calling 'compute'.
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:

-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, 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, 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:

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, 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, 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:

-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, 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, 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:

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

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:

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, etc.

Computing result:

10 * f(x[i-2]) - 140 * f(x[i-1]) - 329 * f(x[i]) + 700 * f(x[i+1]) - 350 * f(x[i+2]) + 140 * f(x[i+3]) - 35 * f(x[i+4]) + 4 * f(x[i+5]) =
420 * f'(x[i]) * h + 5/2 * f^(8)(x[i]) * h^8 + 10/3 * f^(9)(x[i]) * h^9 + 17/6 * f^(10)(x[i]) * h^10 + 20/11 * f^(11)(x[i]) * h^11 + ...

The exact formula:

f'(x[i]) = ( 10 * f(x[i-2]) - 140 * f(x[i-1]) - 329 * f(x[i]) + 700 * f(x[i+1]) - 350 * f(x[i+2]) + 140 * f(x[i+3]) - 35 * f(x[i+4]) + 4 * f(x[i+5]) ) / (420 * h) + O(h^7)

Or

f'(x[i]) = ( 1/42 * f(x[i-2]) - 1/3 * f(x[i-1]) - 47/60 * f(x[i]) + 5/3 * f(x[i+1]) - 5/6 * f(x[i+2]) + 1/3 * f(x[i+3]) - 1/12 * f(x[i+4]) + 1/105 * f(x[i+5]) ) / h + O(h^7)

Julia function:

f1stderiv8pte(f, x, i, h) = ( 10 * f(x[i-2]) - 140 * f(x[i-1]) - 329 * f(x[i]) + 700 * f(x[i+1]) - 350 * f(x[i+2]) + 140 * f(x[i+3]) - 35 * f(x[i+4]) + 4 * f(x[i+5]) ) / (420 * h)

Or

f1stderiv8pte1(f, x, i, h) = ( 1/42 * f(x[i-2]) - 1/3 * f(x[i-1]) - 47/60 * f(x[i]) + 5/3 * f(x[i+1]) - 5/6 * f(x[i+2]) + 1/3 * f(x[i+3]) - 1/12 * f(x[i+4]) + 1/105 * f(x[i+5]) ) / h

Or

f1stderiv8ptd(f, x, i, h) = ( 0.023810 * f(x[i-2]) - 0.333333 * f(x[i-1]) - 0.783333 * f(x[i]) + 1.666667 * f(x[i+1]) - 0.833333 * f(x[i+2]) + 0.333333 * f(x[i+3]) - 0.083333 * f(x[i+4]) + 0.009524 * f(x[i+5]) ) / h

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

Computing result:

f(x[i-4]) - 6 * f(x[i-3]) + 14 * f(x[i-2]) - 4 * f(x[i-1]) - 15 * f(x[i]) + 10 * f(x[i+1]) =
12 * f''(x[i]) * h^2 + 13/15 * f^(6)(x[i]) * h^6 - f^(7)(x[i]) * h^7 + 31/42 * f^(8)(x[i]) * h^8 - 5/12 * f^(9)(x[i]) * h^9 + ...

The exact formula:

f''(x[i]) = ( f(x[i-4]) - 6 * f(x[i-3]) + 14 * f(x[i-2]) - 4 * f(x[i-1]) - 15 * f(x[i]) + 10 * f(x[i+1]) ) / (12 * h^2) + O(h^4)

Or

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

Julia function:

f2ndderiv6pte(f, x, i, h) = ( f(x[i-4]) - 6 * f(x[i-3]) + 14 * f(x[i-2]) - 4 * f(x[i-1]) - 15 * f(x[i]) + 10 * f(x[i+1]) ) / (12 * h^2)

Or

f2ndderiv6pte1(f, x, i, h) = ( 1/12 * f(x[i-4]) - 1/2 * f(x[i-3]) + 7/6 * f(x[i-2]) - 1/3 * f(x[i-1]) - 5/4 * f(x[i]) + 5/6 * f(x[i+1]) ) / h^2

Or

f2ndderiv6ptd(f, x, i, h) = ( 0.083333 * f(x[i-4]) - 0.500000 * f(x[i-3]) + 1.166667 * f(x[i-2]) - 0.333333 * f(x[i-1]) - 1.250000 * f(x[i]) + 0.833333 * f(x[i+1]) ) / h^2

***** Error: 5, -3:1 : i = 5, k[1]*coefs[1][5] + k[2]*coefs[2][5] + ... + k[5]*coefs[5][5] != 0
***** Error: 5, -3:1 : Invalid input because at least 6 points are needed for the 5th derivative.

Computing result:

f(x[i-3]) - 4 * f(x[i-2]) + 6 * f(x[i-1]) - 4 * f(x[i]) + f(x[i+1]) =
f^(4)(x[i]) * h^4 - f^(5)(x[i]) * h^5 + 2/3 * f^(6)(x[i]) * h^6 - 1/3 * f^(7)(x[i]) * h^7 + 11/80 * f^(8)(x[i]) * h^8 + ...

***** Error: 5, -2:2 : i = 5, k[1]*coefs[1][5] + k[2]*coefs[2][5] + ... + k[5]*coefs[5][5] != 0
***** Error: 5, -2:2 : Invalid input because at least 6 points are needed for the 5th derivative.

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 + ...

***** Error: 5, 0:4 : i = 5, k[1]*coefs[1][5] + k[2]*coefs[2][5] + ... + k[5]*coefs[5][5] != 0
***** Error: 5, 0:4 : Invalid input because at least 6 points are needed for the 5th derivative.

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

Computing result:

-f(x[i]) + 5 * f(x[i+1]) - 10 * f(x[i+2]) + 10 * f(x[i+3]) - 5 * f(x[i+4]) + f(x[i+5]) =
f^(5)(x[i]) * h^5 + 5/2 * f^(6)(x[i]) * h^6 + 10/3 * f^(7)(x[i]) * h^7 + 25/8 * f^(8)(x[i]) * h^8 + 331/144 * f^(9)(x[i]) * h^9 + ...

The exact formula:

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

Julia function:

f5thderiv6ptforward(f, x, i, h) = ( -f(x[i]) + 5 * f(x[i+1]) - 10 * f(x[i+2]) + 10 * f(x[i+3]) - 5 * f(x[i+4]) + f(x[i+5]) ) / h^5

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

Computing result:

-47 * f(x[i+2]) + 114 * f(x[i+3]) - 93 * f(x[i+4]) + 26 * f(x[i+5]) =
6 * f'(x[i]) * h + 77/2 * f^(4)(x[i]) * h^4 + 509/5 * f^(5)(x[i]) * h^5 + 1757/12 * f^(6)(x[i]) * h^6 + 6257/42 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'(x[i]) = ( -47 * f(x[i+2]) + 114 * f(x[i+3]) - 93 * f(x[i+4]) + 26 * f(x[i+5]) ) / (6 * h) + O(h^3)

Or

f'(x[i]) = ( -47/6 * f(x[i+2]) + 19 * f(x[i+3]) - 31/2 * f(x[i+4]) + 13/3 * f(x[i+5]) ) / h + O(h^3)

Julia function:

f1stderiv4pte(f, x, i, h) = ( -47 * f(x[i+2]) + 114 * f(x[i+3]) - 93 * f(x[i+4]) + 26 * f(x[i+5]) ) / (6 * h)

Or

f1stderiv4pte1(f, x, i, h) = ( -47/6 * f(x[i+2]) + 19 * f(x[i+3]) - 31/2 * f(x[i+4]) + 13/3 * f(x[i+5]) ) / h

Or

f1stderiv4ptd(f, x, i, h) = ( -7.833333 * f(x[i+2]) + 19 * f(x[i+3]) - 15.500000 * f(x[i+4]) + 4.333333 * f(x[i+5]) ) / h

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

Computing result:

-26 * f(x[i-5]) + 93 * f(x[i-4]) - 114 * f(x[i-3]) + 47 * f(x[i-2]) =
6 * f'(x[i]) * h - 77/2 * f^(4)(x[i]) * h^4 + 509/5 * f^(5)(x[i]) * h^5 - 1757/12 * f^(6)(x[i]) * h^6 + 6257/42 * f^(7)(x[i]) * h^7 + ...

The exact formula:

f'(x[i]) = ( -26 * f(x[i-5]) + 93 * f(x[i-4]) - 114 * f(x[i-3]) + 47 * f(x[i-2]) ) / (6 * h) + O(h^3)

Or

f'(x[i]) = ( -13/3 * f(x[i-5]) + 31/2 * f(x[i-4]) - 19 * f(x[i-3]) + 47/6 * f(x[i-2]) ) / h + O(h^3)

Julia function:

f1stderiv4pte(f, x, i, h) = ( -26 * f(x[i-5]) + 93 * f(x[i-4]) - 114 * f(x[i-3]) + 47 * f(x[i-2]) ) / (6 * h)

Or

f1stderiv4pte1(f, x, i, h) = ( -13/3 * f(x[i-5]) + 31/2 * f(x[i-4]) - 19 * f(x[i-3]) + 47/6 * f(x[i-2]) ) / h

Or

f1stderiv4ptd(f, x, i, h) = ( -4.333333 * f(x[i-5]) + 15.500000 * f(x[i-4]) - 19 * f(x[i-3]) + 7.833333 * 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:

5 * f(x[i-6]) - 72 * f(x[i-5]) + 495 * f(x[i-4]) - 2200 * f(x[i-3]) + 7425 * f(x[i-2]) - 23760 * f(x[i-1]) + 23760 * f(x[i+1]) - 7425 * f(x[i+2]) + 2200 * f(x[i+3]) - 495 * f(x[i+4]) + 72 * f(x[i+5]) - 5 * f(x[i+6]) =
27720 * f'(x[i]) * h - 30/13 * f^(13)(x[i]) * h^13 - f^(15)(x[i]) * h^15 - 29/136 * f^(17)(x[i]) * h^17 - 569/19152 * f^(19)(x[i]) * h^19 + ...

The exact formula:

f'(x[i]) = ( 5 * f(x[i-6]) - 72 * f(x[i-5]) + 495 * f(x[i-4]) - 2200 * f(x[i-3]) + 7425 * f(x[i-2]) - 23760 * f(x[i-1]) + 23760 * f(x[i+1]) - 7425 * f(x[i+2]) + 2200 * f(x[i+3]) - 495 * f(x[i+4]) + 72 * f(x[i+5]) - 5 * f(x[i+6]) ) / (27720 * h) + O(h^12)

Or

f'(x[i]) = ( 1/5544 * f(x[i-6]) - 1/385 * f(x[i-5]) + 1/56 * f(x[i-4]) - 5/63 * f(x[i-3]) + 15/56 * f(x[i-2]) - 6/7 * f(x[i-1]) + 6/7 * f(x[i+1]) - 15/56 * f(x[i+2]) + 5/63 * f(x[i+3]) - 1/56 * f(x[i+4]) + 1/385 * f(x[i+5]) - 1/5544 * f(x[i+6]) ) / h + O(h^12)

Julia function:

f1stderiv12ptcentrale(f, x, i, h) = ( 5 * f(x[i-6]) - 72 * f(x[i-5]) + 495 * f(x[i-4]) - 2200 * f(x[i-3]) + 7425 * f(x[i-2]) - 23760 * f(x[i-1]) + 23760 * f(x[i+1]) - 7425 * f(x[i+2]) + 2200 * f(x[i+3]) - 495 * f(x[i+4]) + 72 * f(x[i+5]) - 5 * f(x[i+6]) ) / (27720 * h)

Or

f1stderiv12ptcentrale1(f, x, i, h) = ( 1/5544 * f(x[i-6]) - 1/385 * f(x[i-5]) + 1/56 * f(x[i-4]) - 5/63 * f(x[i-3]) + 15/56 * f(x[i-2]) - 6/7 * f(x[i-1]) + 6/7 * f(x[i+1]) - 15/56 * f(x[i+2]) + 5/63 * f(x[i+3]) - 1/56 * f(x[i+4]) + 1/385 * f(x[i+5]) - 1/5544 * f(x[i+6]) ) / h

Or

f1stderiv12ptcentrald(f, x, i, h) = ( 0.000180 * f(x[i-6]) - 0.002597 * f(x[i-5]) + 0.017857 * f(x[i-4]) - 0.079365 * f(x[i-3]) + 0.267857 * f(x[i-2]) - 0.857143 * f(x[i-1]) + 0.857143 * f(x[i+1]) - 0.267857 * f(x[i+2]) + 0.079365 * f(x[i+3]) - 0.017857 * f(x[i+4]) + 0.002597 * f(x[i+5]) - 0.000180 * f(x[i+6]) ) / h

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

Computing result:

8 * f(x[i-5]) - 125 * f(x[i-4]) + 1000 * f(x[i-3]) - 6000 * f(x[i-2]) + 42000 * f(x[i-1]) - 73766 * f(x[i]) + 42000 * f(x[i+1]) - 6000 * f(x[i+2]) + 1000 * f(x[i+3]) - 125 * f(x[i+4]) + 8 * f(x[i+5]) =
25200 * f''(x[i]) * h^2 + 50/33 * f^(12)(x[i]) * h^12 + 125/273 * f^(14)(x[i]) * h^14 + 5/72 * f^(16)(x[i]) * h^16 + 1075/154224 * f^(18)(x[i]) * h^18 + ...

The exact formula:

f''(x[i]) = ( 8 * f(x[i-5]) - 125 * f(x[i-4]) + 1000 * f(x[i-3]) - 6000 * f(x[i-2]) + 42000 * f(x[i-1]) - 73766 * f(x[i]) + 42000 * f(x[i+1]) - 6000 * f(x[i+2]) + 1000 * f(x[i+3]) - 125 * f(x[i+4]) + 8 * f(x[i+5]) ) / (25200 * h^2) + O(h^10)

Or

f''(x[i]) = ( 1/3150 * f(x[i-5]) - 5/1008 * f(x[i-4]) + 5/126 * f(x[i-3]) - 5/21 * f(x[i-2]) + 5/3 * f(x[i-1]) - 5269/1800 * f(x[i]) + 5/3 * f(x[i+1]) - 5/21 * f(x[i+2]) + 5/126 * f(x[i+3]) - 5/1008 * f(x[i+4]) + 1/3150 * f(x[i+5]) ) / h^2 + O(h^10)

Julia function:

f2ndderiv11ptcentrale(f, x, i, h) = ( 8 * f(x[i-5]) - 125 * f(x[i-4]) + 1000 * f(x[i-3]) - 6000 * f(x[i-2]) + 42000 * f(x[i-1]) - 73766 * f(x[i]) + 42000 * f(x[i+1]) - 6000 * f(x[i+2]) + 1000 * f(x[i+3]) - 125 * f(x[i+4]) + 8 * f(x[i+5]) ) / (25200 * h^2)

Or

f2ndderiv11ptcentrale1(f, x, i, h) = ( 1/3150 * f(x[i-5]) - 5/1008 * f(x[i-4]) + 5/126 * f(x[i-3]) - 5/21 * f(x[i-2]) + 5/3 * f(x[i-1]) - 5269/1800 * f(x[i]) + 5/3 * f(x[i+1]) - 5/21 * f(x[i+2]) + 5/126 * f(x[i+3]) - 5/1008 * f(x[i+4]) + 1/3150 * f(x[i+5]) ) / h^2

Or

f2ndderiv11ptcentrald(f, x, i, h) = ( 0.000317 * f(x[i-5]) - 0.004960 * f(x[i-4]) + 0.039683 * f(x[i-3]) - 0.238095 * f(x[i-2]) + 1.666667 * f(x[i-1]) - 2.927222 * f(x[i]) + 1.666667 * f(x[i+1]) - 0.238095 * f(x[i+2]) + 0.039683 * f(x[i+3]) - 0.004960 * f(x[i+4]) + 0.000317 * f(x[i+5]) ) / h^2

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

Computing result:

-479 * f(x[i-6]) + 6840 * f(x[i-5]) - 46296 * f(x[i-4]) + 198760 * f(x[i-3]) - 603315 * f(x[i-2]) + 764208 * f(x[i-1]) - 764208 * f(x[i+1]) + 603315 * f(x[i+2]) - 198760 * f(x[i+3]) + 46296 * f(x[i+4]) - 6840 * f(x[i+5]) + 479 * f(x[i+6]) =
302400 * f'''(x[i]) * h^3 + 2478/11 * f^(13)(x[i]) * h^13 + 44089/455 * f^(15)(x[i]) * h^15 + 823/40 * f^(17)(x[i]) * h^17 + 34999/12240 * f^(19)(x[i]) * h^19 + ...

The exact formula:

f'''(x[i]) = ( -479 * f(x[i-6]) + 6840 * f(x[i-5]) - 46296 * f(x[i-4]) + 198760 * f(x[i-3]) - 603315 * f(x[i-2]) + 764208 * f(x[i-1]) - 764208 * f(x[i+1]) + 603315 * f(x[i+2]) - 198760 * f(x[i+3]) + 46296 * f(x[i+4]) - 6840 * f(x[i+5]) + 479 * f(x[i+6]) ) / (302400 * h^3) + O(h^10)

Or

f'''(x[i]) = ( -479/302400 * f(x[i-6]) + 19/840 * f(x[i-5]) - 643/4200 * f(x[i-4]) + 4969/7560 * f(x[i-3]) - 4469/2240 * f(x[i-2]) + 1769/700 * f(x[i-1]) - 1769/700 * f(x[i+1]) + 4469/2240 * f(x[i+2]) - 4969/7560 * f(x[i+3]) + 643/4200 * f(x[i+4]) - 19/840 * f(x[i+5]) + 479/302400 * f(x[i+6]) ) / h^3 + O(h^10)

Julia function:

f3rdderiv12ptcentrale(f, x, i, h) = ( -479 * f(x[i-6]) + 6840 * f(x[i-5]) - 46296 * f(x[i-4]) + 198760 * f(x[i-3]) - 603315 * f(x[i-2]) + 764208 * f(x[i-1]) - 764208 * f(x[i+1]) + 603315 * f(x[i+2]) - 198760 * f(x[i+3]) + 46296 * f(x[i+4]) - 6840 * f(x[i+5]) + 479 * f(x[i+6]) ) / (302400 * h^3)

Or

f3rdderiv12ptcentrale1(f, x, i, h) = ( -479/302400 * f(x[i-6]) + 19/840 * f(x[i-5]) - 643/4200 * f(x[i-4]) + 4969/7560 * f(x[i-3]) - 4469/2240 * f(x[i-2]) + 1769/700 * f(x[i-1]) - 1769/700 * f(x[i+1]) + 4469/2240 * f(x[i+2]) - 4969/7560 * f(x[i+3]) + 643/4200 * f(x[i+4]) - 19/840 * f(x[i+5]) + 479/302400 * f(x[i+6]) ) / h^3

Or

f3rdderiv12ptcentrald(f, x, i, h) = ( -0.001584 * f(x[i-6]) + 0.022619 * f(x[i-5]) - 0.153095 * f(x[i-4]) + 0.657275 * f(x[i-3]) - 1.995089 * f(x[i-2]) + 2.527143 * f(x[i-1]) - 2.527143 * f(x[i+1]) + 1.995089 * f(x[i+2]) - 0.657275 * f(x[i+3]) + 0.153095 * f(x[i+4]) - 0.022619 * f(x[i+5]) + 0.001584 * f(x[i+6]) ) / 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-5]) + 13 * f(x[i-4]) - 69 * f(x[i-3]) + 204 * f(x[i-2]) - 378 * f(x[i-1]) + 462 * f(x[i]) - 378 * f(x[i+1]) + 204 * f(x[i+2]) - 69 * f(x[i+3]) + 13 * f(x[i+4]) - f(x[i+5]) =
3 * f^(8)(x[i]) * h^8 - 31/120 * f^(12)(x[i]) * h^12 - 17/252 * f^(14)(x[i]) * h^14 - 61/6400 * f^(16)(x[i]) * h^16 - 919/997920 * f^(18)(x[i]) * h^18 + ...

The exact formula:

f^(8)(x[i]) = ( -f(x[i-5]) + 13 * f(x[i-4]) - 69 * f(x[i-3]) + 204 * f(x[i-2]) - 378 * f(x[i-1]) + 462 * f(x[i]) - 378 * f(x[i+1]) + 204 * f(x[i+2]) - 69 * f(x[i+3]) + 13 * f(x[i+4]) - f(x[i+5]) ) / (3 * h^8) + O(h^4)

Or

f^(8)(x[i]) = ( -1/3 * f(x[i-5]) + 13/3 * f(x[i-4]) - 23 * f(x[i-3]) + 68 * f(x[i-2]) - 126 * f(x[i-1]) + 154 * f(x[i]) - 126 * f(x[i+1]) + 68 * f(x[i+2]) - 23 * f(x[i+3]) + 13/3 * f(x[i+4]) - 1/3 * f(x[i+5]) ) / h^8 + O(h^4)

Julia function:

f8thderiv11ptcentrale(f, x, i, h) = ( -f(x[i-5]) + 13 * f(x[i-4]) - 69 * f(x[i-3]) + 204 * f(x[i-2]) - 378 * f(x[i-1]) + 462 * f(x[i]) - 378 * f(x[i+1]) + 204 * f(x[i+2]) - 69 * f(x[i+3]) + 13 * f(x[i+4]) - f(x[i+5]) ) / (3 * h^8)

Or

f8thderiv11ptcentrale1(f, x, i, h) = ( -1/3 * f(x[i-5]) + 13/3 * f(x[i-4]) - 23 * f(x[i-3]) + 68 * f(x[i-2]) - 126 * f(x[i-1]) + 154 * f(x[i]) - 126 * f(x[i+1]) + 68 * f(x[i+2]) - 23 * f(x[i+3]) + 13/3 * f(x[i+4]) - 1/3 * f(x[i+5]) ) / h^8

Or

f8thderiv11ptcentrald(f, x, i, h) = ( -0.333333 * f(x[i-5]) + 4.333333 * f(x[i-4]) - 23 * f(x[i-3]) + 68 * f(x[i-2]) - 126 * f(x[i-1]) + 154 * f(x[i]) - 126 * f(x[i+1]) + 68 * f(x[i+2]) - 23 * f(x[i+3]) + 4.333333 * f(x[i+4]) - 0.333333 * f(x[i+5]) ) / h^8

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

Computing result:

f(x[i-5]) - 10 * f(x[i-4]) + 45 * f(x[i-3]) - 120 * f(x[i-2]) + 210 * f(x[i-1]) - 252 * f(x[i]) + 210 * f(x[i+1]) - 120 * f(x[i+2]) + 45 * f(x[i+3]) - 10 * f(x[i+4]) + f(x[i+5]) =
f^(10)(x[i]) * h^10 + 5/12 * f^(12)(x[i]) * h^12 + 1/12 * f^(14)(x[i]) * h^14 + 43/4032 * f^(16)(x[i]) * h^16 + 713/725760 * f^(18)(x[i]) * h^18 + ...

The exact formula:

f^(10)(x[i]) = ( f(x[i-5]) - 10 * f(x[i-4]) + 45 * f(x[i-3]) - 120 * f(x[i-2]) + 210 * f(x[i-1]) - 252 * f(x[i]) + 210 * f(x[i+1]) - 120 * f(x[i+2]) + 45 * f(x[i+3]) - 10 * f(x[i+4]) + f(x[i+5]) ) / h^10 + O(h^2)

Julia function:

f10thderiv11ptcentral(f, x, i, h) = ( f(x[i-5]) - 10 * f(x[i-4]) + 45 * f(x[i-3]) - 120 * f(x[i-2]) + 210 * f(x[i-1]) - 252 * f(x[i]) + 210 * f(x[i+1]) - 120 * f(x[i+2]) + 45 * f(x[i+3]) - 10 * f(x[i+4]) + f(x[i+5]) ) / h^10

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

Computing result:

124 * f(x[i-3]) - 2326 * f(x[i-2]) + 31660 * f(x[i-1]) - 53075 * f(x[i]) + 11640 * f(x[i+1]) + 27132 * f(x[i+2]) - 25928 * f(x[i+3]) + 15990 * f(x[i+4]) - 6900 * f(x[i+5]) + 2010 * f(x[i+6]) - 356 * f(x[i+7]) + 29 * f(x[i+8]) =
25200 * f''(x[i]) * h^2 + 743/33 * f^(12)(x[i]) * h^12 + 50 * f^(13)(x[i]) * h^13 + 35467/546 * f^(14)(x[i]) * h^14 + 187/3 * f^(15)(x[i]) * h^15 + ...

The exact formula:

f''(x[i]) = ( 124 * f(x[i-3]) - 2326 * f(x[i-2]) + 31660 * f(x[i-1]) - 53075 * f(x[i]) + 11640 * f(x[i+1]) + 27132 * f(x[i+2]) - 25928 * f(x[i+3]) + 15990 * f(x[i+4]) - 6900 * f(x[i+5]) + 2010 * f(x[i+6]) - 356 * f(x[i+7]) + 29 * f(x[i+8]) ) / (25200 * h^2) + O(h^10)

Or

f''(x[i]) = ( 31/6300 * f(x[i-3]) - 1163/12600 * f(x[i-2]) + 1583/1260 * f(x[i-1]) - 2123/1008 * f(x[i]) + 97/210 * f(x[i+1]) + 323/300 * f(x[i+2]) - 463/450 * f(x[i+3]) + 533/840 * f(x[i+4]) - 23/84 * f(x[i+5]) + 67/840 * f(x[i+6]) - 89/6300 * f(x[i+7]) + 29/25200 * f(x[i+8]) ) / h^2 + O(h^10)

Julia function:

f2ndderiv12pte(f, x, i, h) = ( 124 * f(x[i-3]) - 2326 * f(x[i-2]) + 31660 * f(x[i-1]) - 53075 * f(x[i]) + 11640 * f(x[i+1]) + 27132 * f(x[i+2]) - 25928 * f(x[i+3]) + 15990 * f(x[i+4]) - 6900 * f(x[i+5]) + 2010 * f(x[i+6]) - 356 * f(x[i+7]) + 29 * f(x[i+8]) ) / (25200 * h^2)

Or

f2ndderiv12pte1(f, x, i, h) = ( 31/6300 * f(x[i-3]) - 1163/12600 * f(x[i-2]) + 1583/1260 * f(x[i-1]) - 2123/1008 * f(x[i]) + 97/210 * f(x[i+1]) + 323/300 * f(x[i+2]) - 463/450 * f(x[i+3]) + 533/840 * f(x[i+4]) - 23/84 * f(x[i+5]) + 67/840 * f(x[i+6]) - 89/6300 * f(x[i+7]) + 29/25200 * f(x[i+8]) ) / h^2

Or

f2ndderiv12ptd(f, x, i, h) = ( 0.004921 * f(x[i-3]) - 0.092302 * f(x[i-2]) + 1.256349 * f(x[i-1]) - 2.106151 * f(x[i]) + 0.461905 * f(x[i+1]) + 1.076667 * f(x[i+2]) - 1.028889 * f(x[i+3]) + 0.634524 * f(x[i+4]) - 0.273810 * f(x[i+5]) + 0.079762 * f(x[i+6]) - 0.014127 * f(x[i+7]) + 0.001151 * f(x[i+8]) ) / h^2

