Monotone cubic interpolation

In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. δ k = y k + 1 − y k x k + 1 − x k {displaystyle delta _{k}={frac {y_{k+1}-y_{k}}{x_{k+1}-x_{k}}}} m k = δ k − 1 + δ k 2 {displaystyle m_{k}={frac {delta _{k-1}+delta _{k}}{2}}} m 1 = δ 1  and  m n = δ n − 1 {displaystyle m_{1}=delta _{1}quad { ext{ and }}quad m_{n}=delta _{n-1},} . α k = m k / δ k  and  β k = m k + 1 / δ k {displaystyle alpha _{k}=m_{k}/delta _{k}quad { ext{ and }}quad eta _{k}=m_{k+1}/delta _{k}} . ϕ k = α k − ( 2 α k + β k − 3 ) 2 3 ( α k + β k − 2 ) > 0 {displaystyle phi _{k}=alpha _{k}-{frac {(2alpha _{k}+eta _{k}-3)^{2}}{3(alpha _{k}+eta _{k}-2)}}>0,} , or τ k = 3 α k 2 + β k 2 {displaystyle au _{k}={frac {3}{sqrt {alpha _{k}^{2}+eta _{k}^{2}}}},} , m k = τ k α k δ k  and  m k + 1 = τ k β k δ k {displaystyle m_{k}= au _{k},alpha _{k},delta _{k}quad { ext{ and }}quad m_{k+1}= au _{k},eta _{k},delta _{k},} . In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i {displaystyle m_{i}} modified to ensure the monotonicity of the resulting Hermite spline. An algorithm is also available for monotone quintic Hermite interpolation.