Texas Instruments TI-89 Tip List page 261

spli4in() does not check xa and xb to ensure that they are within splice interval. Results may be
inaccurate when integrating beyond the interval limits.
Solving for the splice inverse
Some applications require finding x such that s(x)=y, given y. This can only be done if the splice is
strictly monotonic on the splice interval, that is, the splice s(x) has no minimum or maximum on the
interval.
If the splice is monotonic, finding the inverse is a simple matter of using nSolve() as shown in the
function below. We solve the scaled splice function, then un-scale the returned x
are using the scaled splice function, we know that the solution x
we use those as the solution bounds.
spli4x(y,x2,h,cl)
Func
©(s(x),x2,k,{list}) 4th-order splice inverse
©9apr02/dburkett@infinet.com
© Input arguments:
© y value of s(x) at which to find x
© x2 splice interval midpoint
© h splice interval half-width
© cl list of splice coefficients
(nSolve(polyEval(cl,xs)=y,xs)|xs≥⁻1 and xs≤1)*h+x2
EndFunc
This method is satisfactory if you only need a few values of the inverse. There is a faster method to
find many values: find an inverse fourth-order splice to calculate x directly, given f(x). This is simple
once we have found the original splice for y = f(x). We can use splice4() to find the inverse splice; we
just exchange the x's for y's and correct the end-point derivatives:
splice4(xx1,xx2,xx3,xd1,xd2)
where xx1 and xx3 are the interval bounds, as before. For the scaling to work correctly, however,
s(xx2) must be be halfway between yy1 and yy3, so we find yy2 with
yy1 = f 1 ( xx1 )
Given yy2, we solve for the corresponding xx2 with spli4x() above:
spli4x(yy2,xx2x,hx,spl4x)→xx2
but note that xx2x, hx, and spl4x are the values for the original splice, that is, y = s(x). Find they y-axis
half-width hy with
abs(yy2-yy1)→hy
then you can find the inverse derivatives with
hy*((f1(x),x)|x=xx1)→xd1
hy*((f2(x),x)|x=xx3)→xd2
yy3 = f 2 ( xx3 )
is greater than -1 and less than 1, so
s
yy1+yy3
yy2 =
2
to find x. Since we
s
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