TY - JOUR

T1 - Convergence of quartic interpolation splines

AU - Volkov, Yuriy Stepanovich

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The problem of interpolation by quartic splines according to Marsden's scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in L1-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden's scheme for any function from the class C1 on an arbitrary sequence of grids without any constraints. In Marsden's scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin's scheme (the node grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in L1-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class C3.

AB - The problem of interpolation by quartic splines according to Marsden's scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in L1-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden's scheme for any function from the class C1 on an arbitrary sequence of grids without any constraints. In Marsden's scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin's scheme (the node grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in L1-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class C3.

KW - Convergence

KW - Diagonally dominant matrices

KW - Interpolation

KW - Quartic splines

KW - interpolation

KW - convergence

KW - quartic splines

KW - diagonally dominant matrices

UR - http://www.scopus.com/inward/record.url?scp=85078477623&partnerID=8YFLogxK

U2 - 10.21538/0134-4889-2019-25-2-67-74

DO - 10.21538/0134-4889-2019-25-2-67-74

M3 - Article

AN - SCOPUS:85078477623

VL - 25

SP - 67

EP - 74

JO - Trudy Instituta Matematiki i Mekhaniki UrO RAN

JF - Trudy Instituta Matematiki i Mekhaniki UrO RAN

SN - 0134-4889

IS - 2

ER -