Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Numerical techniques based on piecewise polynomial (that is, spline) collation at Gaussian points are exceedingly effective for the approximate solution of boundary value problems, both for ordinary differential equations and for time dependent partial differential equations. There are several widely available computer codes based on this approach, all of which have at their core a particular choice of basis representation for the piecewise polynomials used to approximate the solutions. Until recently, the most popular approach was to use a B-spline representation, but it has been shown that the B-spline basis is inferior, both in operation counts and conditioning, to a certain monomial basis, and the latter has come more into favor. ,  describe a linear algebraic equations which arise in spline collocation at Gaussian points with such a monomial basis. It is shown that the new package, which implements an alternate column and row pivoting algorithm, is a distinct improvement over existing packages from the points of view of speed and storage requirements. In addition,  describe a second package, an important special case of the first, for solving the almost block diagonal systems which arise when condensation is applied to the systems arising in spline collocation at Gaussian points, and also in other methods for solving two-point boundary value problems, such as implicit Runge-Kutta methods and multiple shooting.
['Gouad Majaess', 'Patrick Keast', 'Graeme Fairweather', 'Karin Remington Bennett']
I am writing a research paper on a idea The solution of almost block diagonal linear systems arising in spline collocation at Gaussian points with monomial basis functions
[{"from": "human", "value": "I am writing a research paper on a idea The solution of almost block diagonal linear systems arising in spline collocation at Gaussian points with monomial basis functions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Numerical techniques based on piecewise polynomial (that is, spline) collation at Gaussian points are exceedingly effective for the approximate solution of boundary value problems, both for ordinary differential equations and for time dependent partial differential equations. There are several widely available computer codes based on this approach, all of which have at their core a particular choice of basis representation for the piecewise polynomials used to approximate the solutions. Until recently, the most popular approach was to use a B-spline representation, but it has been shown that the B-spline basis is inferior, both in operation counts and conditioning, to a certain monomial basis, and the latter has come more into favor. ,  describe a linear algebraic equations which arise in spline collocation at Gaussian points with such a monomial basis. It is shown that the new package, which implements an alternate column and row pivoting algorithm, is a distinct improvement over existing packages from the points of view of speed and storage requirements. In addition,  describe a second package, an important special case of the first, for solving the almost block diagonal systems which arise when condensation is applied to the systems arising in spline collocation at Gaussian points, and also in other methods for solving two-point boundary value problems, such as implicit Runge-Kutta methods and multiple shooting."}]