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Finite Difference Formulas

Forward Difference Formulas

1

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

2

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

3

( 11 f i + 18 f i + 1 9 f i + 2 + 2 f i + 3 ) / ( 6 h ) + O ( h 3 )

4

( 25 f i + 48 f i + 1 36 f i + 2 + 16 f i + 3 3 f i + 4 ) / ( 12 h ) + O ( h 4 )

5

( 137 f i + 300 f i + 1 300 f i + 2 + 200 f i + 3 75 f i + 4 + 12 f i + 5 ) / ( 60 h ) + O ( h 5 )

6

( 147 f i + 360 f i + 1 450 f i + 2 + 400 f i + 3 225 f i + 4 + 72 f i + 5 10 f i + 6 ) / ( 60 h ) + O ( h 6 )

Backward Difference Formulas

1

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

2

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

3

( 2 f i 3 + 9 f i 2 18 f i 1 + 11 f i ) / ( 6 h ) + O ( h 3 )

4

( 3 f i 4 16 f i 3 + 36 f i 2 48 f i 1 + 25 f i ) / ( 12 h ) + O ( h 4 )

5

( 12 f i 5 + 75 f i 4 200 f i 3 + 300 f i 2 300 f i 1 + 137 f i ) / ( 60 h ) + O ( h 5 )

6

( 10 f i 6 72 f i 5 + 225 f i 4 400 f i 3 + 450 f i 2 360 f i 1 + 147 f i ) / ( 60 h ) + O ( h 6 )

Central Difference Formulas

1

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

2

( f i 2 8 f i 1 + 8 f i + 1 f i + 2 ) / ( 12 h ) + O ( h 4 )

3

( f i 3 + 9 f i 2 45 f i 1 + 45 f i + 1 9 f i + 2 + f i + 3 ) / ( 60 h ) + O ( h 6 )

4

( 3 f i 4 32 f i 3 + 168 f i 2 672 f i 1 + 672 f i + 1 168 f i + 2 + 32 f i + 3 3 f i + 4 ) / ( 840 h ) + O ( h 8 )