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In this subsection, we construct Nordsieck L-stable SGLMs with p = q = s = r - 1 = 4, the abscissa vector c = [[[1/4] [1/2] [3/4] 1].sup.T], the error constant [C.sub.5] = -[10.sup.-5] and IQS property.
Construction of SGLMs with quadratic stability property is desirable.
Numerical results for the SGLMs of order p = 1, 2, 3, 4.
The general form of SGLM for the numerical solution of initial value problem of the form
Here, A, [bar.A] [member of] [R.sup.sxs], U [member of] [R.sup.sxr], B, [bar.B] [member of] [R.sup.rxs], and V [member of] [R.sup.rxr] are six coefficients matrices of the SGLM. Also, p and q are respectively order and stage order of the method, r is the number of input and output approximations, and s is the number of internal stages.
 An SGLM in Nordsieck form has order p and stage order q = P if and only if
A Nordsieck SGLM with p = q = s = r - 1 and coefficients matrices A, [bar.A], B, [bar.B], U, and V defined by (1.2) has IQS property if there exists a matrix X [member of] [R.sup.(s+1)x(s+1)] such that
Assume that the SGLM (1.1) with p = q = s = r - 1 has IQS property and that the matrices [I.sub.s] - zA - [z.sup.2][bar.A] and [I.sub.s+1] - zX are nonsingular.
For an SGLM with p = q = s = r - 1, the general form of matrix X satisfying IQS conditions is
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