Numeriska beräkningar i Naturvetenskap och Teknik 1.Solving equations.

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Presentation transcript:

Numeriska beräkningar i Naturvetenskap och Teknik 1.Solving equations

Numeriska beräkningar i Naturvetenskap och Teknik Discretization

Numeriska beräkningar i Naturvetenskap och Teknik Error propagation

Numeriska beräkningar i Naturvetenskap och Teknik Discretization and error propagation

Numeriska beräkningar i Naturvetenskap och Teknik An example using graphs

Numeriska beräkningar i Naturvetenskap och Teknik A numerical exemple >> bisection method Sign change Half of the interval... again... and again...

Numeriska beräkningar i Naturvetenskap och Teknik Bisection

Numeriska beräkningar i Naturvetenskap och Teknik Bisection method A clear limitation of the method is that the new approximation does not take into account the value of the function for the latest x-value it was calculated at. compare which gives So, we have stepped two times but are hardly any closer to the solution than we were two steps earlier…(the sign has changed though…) How can we use the knowledge we have of the value of the function in order to guess a new better value?

Numeriska beräkningar i Naturvetenskap och Teknik Secant method Sekantens ekvation Iteration formula! Root x 2, approximation

Numeriska beräkningar i Naturvetenskap och Teknik The secant method Code example for the secant method in the diff. equation section

Numeriska beräkningar i Naturvetenskap och Teknik Newton-Raphson’s method Let the step between x n and x n-1 tend to zero... or the eq. of the tangent…

Numeriska beräkningar i Naturvetenskap och Teknik Iteration principle 1-point method 2-point method

Numeriska beräkningar i Naturvetenskap och Teknik Our exemple once more: Solve for the root: i.e.is a possibility. Are there others? yes, infinitely many!

Numeriska beräkningar i Naturvetenskap och Teknik

The mean value theorem: but i.e. or

Numeriska beräkningar i Naturvetenskap och Teknik The relative error between two iterations If G’(ξ) is less than 1 the iteration will converge If x n och x n+1 are close to a then ξ is also an approximation for a, i.e. if G’(ξ)<1 around the root a, the iteration will converge. The convergence is quicker the smaler G’ is in the surrounding of a How can this be used in order to optimize how we write the iteration formula?

Numeriska beräkningar i Naturvetenskap och Teknik Let us rewrite f(x)=0 that is Assume that the start value is a good approximation for the root: which leads to

Numeriska beräkningar i Naturvetenskap och Teknik Newton Raphson’s modified method: Our exemples Applying Newton Raphson’s modified method

Numeriska beräkningar i Naturvetenskap och Teknik

Kodexempel, Newton Raphsons modifierade metod etc.