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Finite Difference Formulas

Forward Difference Formulas

2

( f i 2 f i + 1 + f i + 2 ) / h 2 + O ( h )

3

( 2 f i 5 f i + 1 + 4 f i + 2 f i + 3 ) / h 2 + O ( h 2 )

4

( 35 f i 104 f i + 1 + 114 f i + 2 56 f i + 3 + 11 f i + 4 ) / ( 12 h 2 ) + O ( h 3 )

5

( 45 f i 154 f i + 1 + 214 f i + 2 156 f i + 3 + 61 f i + 4 10 f i + 5 ) / ( 12 h 2 ) + O ( h 4 )

6

( 812 f i 3132 f i + 1 + 5265 f i + 2 5080 f i + 3 + 2970 f i + 4 972 f i + 5 + 137 f i + 6 ) / ( 180 h 2 ) + O ( h 5 )

7

( 938 f i 4014 f i + 1 + 7911 f i + 2 9490 f i + 3 + 7380 f i + 4 3618 f i + 5 + 1019 f i + 6 126 f i + 7 ) / ( 180 h 2 ) + O ( h 6 )

Backward Difference Formulas

2

( f i 2 2 f i 1 + f i ) / h 2 + O ( h )

3

( f i 3 + 4 f i 2 5 f i 1 + 2 f i ) / h 2 + O ( h 2 )

4

( 11 f i 4 56 f i 3 + 114 f i 2 104 f i 1 + 35 f i ) / ( 12 h 2 ) + O ( h 3 )

5

( 10 f i 5 + 61 f i 4 156 f i 3 + 214 f i 2 154 f i 1 + 45 f i ) / ( 12 h 2 ) + O ( h 4 )

6

( 137 f i 6 972 f i 5 + 2970 f i 4 5080 f i 3 + 5265 f i 2 3132 f i 1 + 812 f i ) / ( 180 h 2 ) + O ( h 5 )

7

( 126 f i 7 + 1019 f i 6 3618 f i 5 + 7380 f i 4 9490 f i 3 + 7911 f i 2 4014 f i 1 + 938 f i ) / ( 180 h 2 ) + O ( h 6 )

Central Difference Formulas

1

( f i 1 2 f i + f i + 1 ) / h 2 + O ( h 2 )

2

( f i 2 + 16 f i 1 30 f i + 16 f i + 1 f i + 2 ) / ( 12 h 2 ) + O ( h 4 )

3

( 2 f i 3 27 f i 2 + 270 f i 1 490 f i + 270 f i + 1 27 f i + 2 + 2 f i + 3 ) / ( 180 h 2 ) + O ( h 6 )