In [3] we have presented a method for the construction of an approximation to the solution of the following initial-value problem for the first-order Volterra integro-differential equation (VIDE)

with the initial condition $y(0)=y_{0}$, by polynomial spline functions. Here, the given functions $f:I\times R\rightarrow R$ and $K:S\times R\rightarrow R$ (with $S:=\{(t,s):0\leq s\leq t\leq T\}$ ) are supposed to be sufficiently smooth for the initial-value problem for VIDE (1.1) to have a unique solution $y\in C^{\alpha}(I)$, with $\alpha\in\mathbf{N}$ (see [6]).

In order to describe this method, let $\Pi_{N}:0=t_{0}<t_{1}<\ldots<i_{N}=T$ (with $t_{n}=t_{n}^{(N)}$ ) be a quasi-uniform mesh for the given interval $I$, and set

Moreover, let $\mathscr{P}_{k}$ denote the space of (real) polynomials of a degree not exceeding $k$. Then we define, for given integers $m$ and $d$ with $m\geq 1$ and $d\geq-1$,

to be the space of polynomial splines of degree $m+d$, whose elements possess the knots $Z_{N}$ and are $d$-times continually differentiable on $I$. If $d=-1$, then the elements of $S_{m-1}^{(-1)}\left(Z_{N}\right)$ may have jump discontinuities at the knots $Z_{N}$.

An element $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ has for all $n=0,1,\ldots,N-1$ and for all $t\in\sigma_{n}$ the following form (see [7])

From (1.2) we see that an element $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ is well defined when we know the coefficients $\left\{a_{n,r}\right\}_{r=\overline{1,m}}$ for all $n=0,\ldots,N-1$. In order to determine these coefficients, we consider the set of collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, where $0<c_{1}<<\ldots<c_{m}\leq 1$, and we define the set of collocation points by

The approximate solution $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ will be determined by imposing the condition that $u$ satisfies the initial-value problem (1.1) on $X(N)$

The above algorithm determines a unique approximate solution $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ whose convergence and local superconvergence properties have been studied in [3].

In this paper, we will analyze the numerical stability of the polynomial spline collocation method in the case in which the mesh sequences $\left\{\Pi_{N}\right\}_{N}$ are uniform, i.e., $h_{n}=h$, for all $n=0,1,\ldots,N-1$.

In order to discuss numerical stability, we study the behavior of the method as applied to the Volterra integro-differential equation

with the initial condition $y(0)=y_{0}$. Here, the given function $f:I\rightarrow R$ is supposed to be sufficiently smooth (i.e., $f\in C^{\alpha}(I)$, with $\alpha\geq 1$ ).

This equation is called the basis test equation and it was suggested by Brunner and Lambert in 1974 (see [2]), and then it has been extensively used for investigating stability properties of several methods.

Henceforward, we will refer to a polynomial spline collocation method in the space $S_{m+d}^{(d)}\left(Z_{N}\right)$, simply as an ( $m,d$ )-method (see [4], [5]).

Definition 2.1. An ( $m,d$ )-method is said to be stable if all solutions $\left\{u\left(t_{n}\right)\right\}$ remain bounded, as $n\rightarrow\infty,h\rightarrow 0$ while hN remains fixed.

From relation (1.2) we observe that the first $d+1$ coefficients of the polynomial $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ are determined by the smooth condition, and the last $m$ coefficients are determined by the collocation conditions. Thus, it is convenient to introduce the following notations:

With these notations, for all $t:=t_{n}+\tau h\in\sigma_{n}$, (1.2) becomes

for all $\tau\in(0,1]$ and $n=0,1,\ldots,N$.
Now, for $d\geq 1$, if we apply the collocation method to test integral equation (2.1) and we use the representation (2.3), we obtain the following collocation equation

where $V$ is the $m\times m$ matrix, $W$ is the $m\times(d+1)$ matrix, and $r_{n}$ is the $m$-vector, whose elements are

and
$r_{n,j}:=\left\{\begin{array}[]{l}f\left(t_{0,j}\right)-f\left(t_{0}\right),\text{ if }n=0,\\ f\left(t_{n,j}\right)-f\left(t_{n-1,m}\right)+u_{n-1}^{\prime}\left(t_{n-1,m}\right)-u_{n-1}^{\prime}\left(t_{n}\right)+\alpha\left[u_{n-1}\left(t_{n}\right)-u_{n-1}\left(t_{n-1,m}\right)\right]+\\ +\lambda h\int_{c_{m}}^{1}u_{n-1}\left(t_{n-1}+\tau h\right),\text{ if }n>0.\end{array}\right.$
By direct differentiation of relations (2.3), for the smooth conditions of the approximation $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$, we get a relation between vector $\eta_{n+1}$ and vectors $\eta_{n}$ and $\beta_{n}$, respectively

where $A$ is the $(d+1)\times(d+1)$ upper triangular matrix, and $B$ is the $(d+1)\times m$ matrix, whose elements are

In order to prove the results concerning the numerical stability properties of the polynomial spline collocation method, we need the following lemma (see [8]):

LEMMA 2.2. For any matrix $P$ and any $\varepsilon>0$, there exists a subordinate norm such that $\|P\|\leq S(P)+\varepsilon$, with $S(P):=\max\left\{\left|\lambda_{j}\right|;\lambda_{j}\right.$ are the eigenvalues of $\left.P\right\}$. If $P$ is of class $M$, then there exists a norm such that $\|P\|=S(P)$.

By means of this lemma we can characterize numerical stability in the terms of eigenvalues of the suitable matrix. The following theorem represents a stability criterion for our method:

THEOREM 2.3. An ( $m$, d)-method is stable if and only if all eigenvalues of matrix $M:=A+BV^{-1}W$ are in the unit disk and all eigenvalues with $|\mu|=1$ belong to $1\times 1$ Jordan block.

Proof. In order to prove this theorem, we will show that the vectors $\eta_{n}$ and $\beta_{n}$, defined by (2.2), are uniformly bounded for $h\searrow 0,n\rightarrow\infty$, while $hN$ remains fixed, i.e., there exist two finite constants $M_{1}$ and $M_{2}$, such that

uniformly in $n$, as $h\searrow 0$. These in turn imply, according to (2.3), that

and from Definition 2.1, it results that an ( $m$, d)-method is stable.
From the form of matrix $V$ we see that for $h$ small enough, this matrix is nonsingular. Elimination of $\beta_{n}$ between (2.4) and (2.5) yields

for all $n=0,1,\ldots,N-2$. Thus, relations (2.4) and (2.6) imply that for all $n=0,1$, $...N-1$, we have

Because the first derivative of the given function $f$ is a continuous function on $I$, it results that there exists a positive constant $L$ such that $\left|f^{\prime}(t)\right|\leq L$, for all $t\in I$; and, for all $n=0,\ldots,N-1$, we have

In the case in which $c_{m}=1$, relation (2.8) becomes

and from relation (2.7) we obtain

Using Lemma 2.2, it results from (2.10) that

From these relations, it results that $\left\|\eta_{n}\right\|_{1}$ and $\left\|\beta_{n}\right\|_{1}$ remain bounded for $n\rightarrow\infty,h\rightarrow 0$ and $Nh=T$, if and only if $S(M)\leq 1$.

In the case in which $c_{m}\neq 1$, then we will prove by induction that relations (2.9) and (2.10) hold if we change the constant $L_{1}$, in (2.9), with a new finite constant $L_{2,n}$, defined by

where

and, respectively, in (2.10) we take

For $n=0$, relations (2.7) become: $\eta_{0}=\eta_{0}$ and $\beta_{0}=V^{-1}W\eta_{0}+V^{-1}r_{0}$, respectively. Because the matrices $V^{-1},W$ and the vector $r_{0}$ are bounded in norm for $h\rightarrow 0$, it results that the vector $\beta_{0}$ is bounded, too. Thus, by the definition relation (2.3), we obtain $\left|u_{0}(\tau h)\right|<\infty$, and $\left|u_{0}^{\prime}(\tau h)\right|<\infty$, for all $\tau\in[0,1]$, hence, by (2.8), it results that $\left\|r_{1}\right\|_{1}\leq hL_{2,1}$, with $L_{2,1}<\infty$.

Now we suppose that $\left\|\beta_{j}\right\|_{1}\leq\infty$ and $\left\|\eta_{j}\right\|_{1}<\infty$, for all $j=0,1,\ldots,n-1$. Under this assumption, by (2.3) it results that $\left|u_{n-1}(t)\right|<\infty$, and $\left|u_{n-1}^{\prime}(t)\right|<\infty$, for all $t\in\sigma_{n-1}$; furthermore, by (2.8) it follows that $\left\|r_{n}\right\|_{1}\leq hL_{2,n}$, with $L_{2,n}<\infty$. Moreover, relations (2.5) and (2.4) imply $\left\|\eta_{n}\right\|_{1}<\infty$, and $\left\|\beta_{n}\right\|_{1}<\infty$, respectively. Thus, using the bound of $r_{n}$, from (2.7) it results that relations (2.10) and (2.11) hold with $L_{1}$ replaced by $L_{2}$, for all $n=0,1,\ldots,N-1$; accordingly, the theorem is fully demonstrated.

Remark 2.4. From (2.6) we see that the dimension of matrix $M$ is $\operatorname{dim}M:=d+1$. Moreover, if we denote by $M_{0}$ the matrix $M$ with $h=0$, and by $\mu^{(0)}$ and $\mu$ the eigenvalues of $M_{0}$ and $M$, respectively, then it follows that the matrix $\mathrm{M}_{0}$ has $\mu_{1}^{(0)}=\mu_{2}^{(0)}=1$, for all $m\geq 0$ and $d\geq 1$.

In the following we will investigate some special cases.
I. $d=1$. In this case the approximation space is $S_{m+1}^{(1)}\left(Z_{N}\right)$. From Theorem 2.3 and Remark 2.4, the following theorem results:

THEOREM 3.1. An ( $m,0$ )-method is stable for all $m\geq 1$, and for every choice of the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1},m}$.

The above theorem may be directly proved by using the same technique as in the first application from [4].
II. $m=1$. This choice of $m$ corresponds to a classical spline function, i.e., $u\in S_{d+1}^{(d)}\left(Z_{N}\right),d\geq 1$. Using notations from Remark 2.4 (ie., $M_{0}$ is the matrix $M$, with $h=0$, and by $\mu^{(0)}$ and $\mu$, the eigenvalues of $M_{0}$ and $M$, respectively), we have

If $c_{1}\in(0,1]$ is the collocation parameter, then, for all $d\geq 1$, using the binomial expansion, we find for matrix $M_{0}$ the trace

As regards the stability of the spline collocation method, we have the following result:

THEOREM 3.2. A $(1,d)$-method is stable if and only if one from the following conditions is true:
(i) $d=1$ and $c_{1}\in(0,1]$;
(ii) $d=2$ and $c_{1}=1$.

Proof. In the case $d=1$, this theorem follows from Theorem 3.1. If $d=2$, then the third eigenvalue of $M_{0}$ is $\mu_{3}^{(0)}=1-\frac{2}{c_{1}}\leq-1$, for $c_{1}\in(0,1]$, and its absolute value is 1 , if and only if $c_{1}=1$. For $d\geq 3$, from relation (3.1), we obtain

and $\mu_{2}^{(0)}+\mu_{3}^{(0)}\leq-4$, if $d=2$. Since $\operatorname{Tr}\left(M_{0}\right)=\mu_{1}^{(0)}+\mu_{2}^{(0)}+\ldots+\mu_{d+1}^{(0)}<-(d+1)$, it results that there exists an eigenvalue $\mu^{(0)}$ whose value is smaller than -1 , i.e., $\left|\mu^{(0)}\right|>1$. Thus, from Theorem 2.3 we have that, for $d\geq 2,\mathrm{a}(1,d)$-method is unstable for every choice of the collocation parameter $c_{1}\in(0,1]$.
III. $m=2$. In this case, we find for the trace of matrix $M_{0}$ the relation

where $0<c_{1}<c_{2}\leq 1$ are the collocation parameters. Using the above relation, we obtain the following

THEOREM 3.3. (i) $A(2,1)$-method is stable for every choice of the collocation parameters.
(ii) A(2,2)-method is stable if and only if $c_{1}+c_{2}\geq\frac{3}{2}$.
(iii) If $c_{2}=1$, then $a(2,d)$-method is unstable for all $d\geq 3$.

Proof. Assertion (i) follows from Theorem 3.1. To prove assertion (ii), it is enough to observe that, for $d=1$, the third eigenvalue of $M_{0}$ is $\mu_{3}^{(0)}=\frac{c_{1}c_{2}-2\left(c_{1}+c_{3}\right)+3}{c_{1}c_{2}}$, and the stability condition $\left|\mu_{2}^{0}\right|\leq 1$ is equivalent to the condition $c_{1}+c_{2}\geq\frac{3}{2}$.

If $d=2$ and $c_{2}=1$, then one of the eigenvalues of $M_{0}$ is

here we have $\mu_{3}^{(0)}>1$ for every choice of the collocation parameter $c_{1}\in(0,1)$. Thus, assertion (iii) holds for $d=3$. If $d>3$, then for $c_{2}=1$ the formula (3.2) becomes

for all $c_{1}\in(0,1)$, and thus the assertion of this theorem follows from Theorem 2.3.
IV. $d=2$. In this case, approximation $u\in S_{m+2}^{(2)}\left(Z_{N}\right)$, the dimension of the matrix $M_{0}$ is 3 , and $\mu_{1}^{(0)}=\mu_{2}^{(0)}=1$ are its first two eigenvalues. By direct computation, for the third eigenvalue of $M_{0}$, we find

where

In view of the results obtained for $m=1,2,\ldots,6$ we are led to the following affirmation:

Conjecture 3.4. If $d=2$, then the third eigenvalue of $M_{0}$ may be calculated by using relation (3.3) for all $m\geq 1$.

Now, if we denote by $R_{m}(t)$ the polynomial of degree $m$ whose zeros are the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$, then we have the following stability criterion:

Theorem 3.5. An ( $m,2$ )-method is stable if and only if

Proof. Since $R_{m}(t)$ is the polynomial of degree $m$ whose zeros are the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, using notation (3.4), it may be written

Thus, from (3.3), (3.5) and (3.6), we obtain

and so, if Conjecture 3.4 is true, the assertion of this theorem follows from Theorem 2.3.

COROLLARY 3.6. If the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$ are uniformly distributed in $(0,1]\left(i.e.,c_{j}:=\frac{j}{m}\right.$, for all $\left.j=1,2,\ldots,m\right)$, then an $(m,2)$ method is stable.

If Conjecture 3.4 is true, then for $c_{m}=1$ the above conjecture and theorem become

COROLLARY 3.7. If the last collocation parameter is one (i.e., $c_{m}=1$ ), then:
(i) The third eigenvalue of $M_{0}$ may be calculated by using the relation

for all $m\geq 1$, where

(ii) An (m, 2)-method is stable if and only if

where $R_{m}(t)$ is the polynomial of degree $m$ defined by (3.8).
Proof. (i) If the last collocation parameter is one (i.e., $c_{m}=1$ ), then, from

where $S_{k}^{\prime}$ are defined in (3.8). Now, the first assertion of this corollary follows by Conjecture 3.4 and relation (3.10).
(ii) Using notations (3.8), the polynomial $R_{m}$, whose zeros are the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, may be written

Thus, from (3.7), (3.9) and (3.11), we obtain

and so, the second assertion of this corollary follows from Theorem 2.3.
In [3] we have proved that, in a suitable choice of the collocation parameters, we obtain an approximated solution which has a local convergence order greater than the global order, in the points from $Z_{N}$. As regards the stability of this local superconvergent solution $u\in S_{m+2}^{(2)}\left(Z_{N}\right)$, we have

COROLLARY 3.8. (i) If the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$ are the Radau II points from ( 0,1 ], then an ( $m,2$ )-method is unstable for all $m\geq 2$.
(ii) If the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$ are the Gauss points from $(0,1)$, then an ( $m,2$ )-method is unstable for all $m\geq 2$.
(iii) If the first $m-1$ collocation parameters $\left\{c_{j}\right\}_{j=1,m}$ are the Gauss points from $(0,1)$, and the last is $c_{m}=1$, then an ( $m,2$ )-method is stable for all $m\geq 2$.

Proof. The results from this corollary follow from assertion (ii) of Corollary 3.7 and the properties of the Radau II points and Gauss points, respectively. In this proof we will denote by $P_{m}(s)$ the Legendre’s polynomial of a degree not expanding $m$, for $s\in[-1,1]$.
(i) If the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$ are the Radau II points from $(0,1]$, then the polynomial $R_{m}$, whose zeros are the collocation parameters $\left\{c_{j}\right\}_{j=\mathrm{r},\bar{m}}$, may be written

Thus, using the properties of Legendre’s polynomial, from (3.9), we obtain

(ii) If the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$ are the Gauss points from $(0,1)$, then the polynomial $R_{m}$ is

Because $P^{\prime}{}_{m}(1)=\frac{m(m+1)}{2}$, from (3.9) it results that $\left|\mu_{2}^{(0)^{\prime}}\right|=m(m+1)>1$, for all $m\geq 2$.
(iii) In this choice of collocation parameters, polynomial $R_{m}$ becomes

and, from (3.9), we obtain

for all $m\geq{}^{A}1$.
V. $d=0$. In the end of this section we analyze the numerical stability of the spline collocation method in the space $S_{m}{}^{(0)}$, for $m\geq 1$. An element $u\in S_{m}^{(0)}\left(Z_{N}\right)$ has for all $n=0,1,\ldots,N-1$ the form