Math Editor

Math 5 Interpolation

Lagrange's Formula

PL4​(x)​=L0​⋅y0​+L1​⋅y1​+L2​⋅y2​+L3​⋅y3​+L4​⋅y4​=(1−2)(1−3)(1−4)(1−5)(x−2)(x−3)(x−4)(x−5)​×0.25+(2−1)(2−3)(2−4)(2−5)(x−1)(x−3)(x−4)(x−5)​×0.17+(3−1)(3−2)(3−4)(3−5)(x−1)(x−2)(x−4)(x−5)​×0.11+(4−1)(4−2)(4−3)(4−5)(x−1)(x−2)(x−3)(x−5)​×0.1+(5−1)(5−2)(5−3)(5−4)(x−1)(x−2)(x−3)(x−4)​×0.08=0.23+0.1325x−0.15125x2+0.0425x3−0.00374997x4​

PL4​(1.5)=0.21289PL4​(3.5)=0.10039

Newton's Divided difference Formula

Divided Difference Table

Forward Divided Difference

starting at x=1

PNFDD4​(x)=0.25−0.08(x−1)+0.01(x−1)(x−2)+0.005(x−1)(x−2)(x−3)−0.00375(x−1)(x−2)(x−3)(x−4)

PNFDD3​(x)=0.25−0.08(x−1)+0.01(x−1)(x−2)+0.005(x−1)(x−2)(x−3)

Backward Divided Difference

starting at x=5

PNBDD4​(x)=0.08−0.02(x−5)−0.005(x−5)(x−4)−0.01(x−5)(x−4)(x−3)−0.00375(x−5)(x−4)(x−3)(x−2)

starting at x=4

PNBDD3​(x)=0.1−0.01(x−4)+0.025(x−4)(x−3)+0.005(x−4)(x−3)(x−2)

Newton's Difference Formula

Difference Table

Forward Difference

starting at x=1

SNFD4​(x)=1x−1​⟹S(1.5)=0.5

PNFD4​(x)=0.25−0.08S+0.02×2!S(S−1)​+0.03×3!S(S−1)(S−2)​−0.09×4!S(S−1)(S−2)(S−3)​

Backward Difference

starting at x=4

SNBD3​(x)=1x−4​⟹S(3.5)=−0.5

PNBD3​(x)=0.1−0.01S+0.05×2!S(S+1)​+0.03×3!S(S+1)(S+2)​

Gauss's Formula

Difference Table

Forward Difference

starting at x=3

SGFD4​(x)=1x−3​⟹S(3.5)=0.5

PGFD4​(x)=0.11−0.01S+0.05×2!S(S−1)​−0.06×3!S(S−1)(S+1)​−0.09×4!S(S−1)(S+1)(S−2)​

Backward Difference

starting at x=3

SGBD4​(x)=1x−3​⟹S(3.5)=0.5

PGBD4​(x)=0.11−0.06S+0.05×2!S(S+1)​+0.03×3!S(S+1)(S−1)​−0.09×4!S(S+1)(S−1)(S+2)​