A. V. Ivanov

ON THE STABILITY OF CERTAIN COMPUTATIONAL ALGORITHMS INVERTING REGULAR DIFFERENCE EQUATIONS

(Presented by Academician V. I. Smirnov, 27 VI 1962)

Let the norm of a vector $y^n$ in an $n$-dimensional vector space be taken to be the norm

\[ \|y^n\|=\max_{i=1,\ldots,n}|y_i^n|, \tag{1} \]

and let $\|l\|$ denote the norm of the matrix $l$, acting as a linear operator in the $n$-dimensional space with norm (1).

Suppose a linear algebraic system is given,

\[ l^n y^n=f^n, \tag{2} \]

depending on its order $n$ as a parameter $(n\geq N,\ N$ a fixed natural number), and suppose the matrices $l^n$ are tridiagonal:

\[ l^n= \left( \begin{array}{ccccccccc} -b_1^n & c_1^n \\ a_2^n & -b_2^n & c_2^n \\ & \cdot & \cdot & \cdot \\ & & a_i^n & -b_i^n & c_i^n \\ & & & \cdot & \cdot & \cdot \\ & & & & a_{n-1}^n & -b_{n-1}^n & c_{n-1}^n \\ & & & & & a_n^n & -b_n^n \end{array} \right). \]

The matrix $l^n$, depending on the parameter $n$, will be called a regular difference operator if there exist positive numbers $\nu$ and $\mu$ $(\nu<\mu)$, independent of $n$, such that for all $n\geq N$

\[ a_i^n,\ c_i^n\geq \nu,\qquad a_i^n+c_i^n\leq b_i^n,\qquad b_i^n\leq \mu,\qquad i=1,2,\ldots,n. \tag{3} \]

In the inequalities (3) and below, by $a_1^n$ and $c_n^n$ we mean the expressions $b_1^n-c_1^n$ and $b_n^n-a_n^n$. Suppose the matrix of system (2) is a regular difference operator. In that case we shall call system (2) a regular difference equation (r. d. e.).

Theorem 1. For every $n\geq N$, the r. d. e. (2) is uniquely solvable.

Let $\omega_1$ and $\omega_2$ be arbitrary nonnegative real numbers. Denote by $\mathfrak{M}(\omega_1,\omega_2)$ the class of all regular difference operators for which the equalities

\[ \min\left\{ \sup_{\substack{n\geq N\\ 1\leq i\leq j\leq n}} \prod_{k=i}^{j}\frac{a_k^n}{c_k^n}, \quad \sup_{\substack{n\geq N\\ 1\leq i\leq j\leq n}} \prod_{k=i}^{j}\frac{c_k^n}{a_k^n} \right\} = \frac{1}{\omega_1}, \qquad \inf_{\substack{n\geq N\\ 1<i<n}} \left(b_i^n-a_i^n-c_i^n\right)=\omega_2. \]

Let \(\omega_3\) be an arbitrary real number. We shall say that the R.D.E. (2) belongs to the class \(\mathfrak M(\omega_1,\omega_2)\times\mathfrak M(\omega_3)\), if the matrix of this system belongs to the class \(\mathfrak M(\omega_1,\omega_2)\) and there exists a positive number \(K\) (independent of \(n\)) such that for all \(n\geqslant N\), for the right-hand side of equation (2), the inequality \(\|\mathbf f^n\|\leqslant K n^{\omega_3}\) holds.

Let \((l^n)^{-1}\) be the matrix inverse to \(l^n\). We shall say that the order of conditioning of the class \(\mathfrak M(\omega_1,\omega_2)\) is equal to \(-\lambda\), if \(\lambda\) is the smallest of all numbers \(p\) such that, for each of the regular difference operators belonging to \(\mathfrak M(\omega_1,\omega_2)\), there is a positive number \(C\) (independent of \(n\)) such that for all \(n\geqslant N\) the inequality
\[ \|(l^n)^{-1}\|\leqslant Cn^p \tag{4} \]
is satisfied.

If inequality (4) cannot be established for any \(p\) for all operators of the class \(\mathfrak M(\omega_1,\omega_2)\), then we shall say that the order of conditioning of the class \(\mathfrak M(\omega_1,\omega_2)\) is equal to \(-\infty\).

Theorem 2. The order of conditioning of the class \(\mathfrak M(\omega_1,\omega_2)\) is expressed by the value of the function \(\lambda(\omega_1,\omega_2)\), which has the form
\[ \lambda(\omega_1,\omega_2)= \begin{cases} -\infty, & \omega_1=0,\quad \omega_2=0,\\ -2, & 0<\omega_1\leqslant 1,\quad \omega_2=0,\\ -1, & \omega_1>1,\quad \omega_2=0,\\ p(\omega_1,\omega_2), & \omega_2>0, \end{cases} \]
where \(p(\omega_1,\omega_2)\geqslant 0\) for \(\omega_1>0\).

Suppose that, in order to invert the R.D.E. (2), some computational algorithm (c.a.) \(Q\) is applied, consisting in specifying, for all \(n\geqslant N\), the system of equations
\[ x_k=Q_k^n(\mathbf x_{k-1},\mathbf F),\qquad k=1,2,\ldots,m_n, \tag{5} \]
where \(\mathbf F=(F_1,F_2,\ldots,F_{r_n})\) is a vector composed of the elements of the matrix \(l^n\) and the components of the right-hand side \(\mathbf f^n\) of the R.D.E. (2); \(\mathbf x_{k-1}=(x_1,x_2,\ldots,x_{k-1})\); \(Q_k^n\) is some arithmetic or logical operation on certain components of the vectors \(\mathbf x_{k-1}\) and \(\mathbf F\). Let the values \(x_{k_i}\) \((i=1,2,\ldots,n)\) represent the values \(y_i^n\) of the solution \(\mathbf y^n\) of the R.D.E. (2). This form of writing a computational algorithm is used in work \((^1)\).

We shall say that the c.a. \(Q\) is weakly stable on the R.D.E. (2), if the corresponding system of equations (5) satisfies the following conditions:

1) there exists a positive number \(C_1\), independent of \(n\), such that for all \(n\geqslant N\)
\[ |x_k|\leqslant C_1,\qquad k=1,2,\ldots,m_n; \tag{6} \]

2) the system of equations (5) can be divided in such a way into \(l\) links
\[ \begin{gathered} x_{jk}=Q_{jk}^n(\mathbf x_{jk-1},\mathbf F_j),\qquad k=1,2,\ldots,m_j;\quad j=1,2,\ldots,l;\\ m_1+m_2+\cdots+m_l=m_n, \end{gathered} \tag{7} \]
where \(\mathbf F_j\) are vectors composed of certain components of the vector \(\mathbf F\) and components of the vectors \(\mathbf x_{sm_s}\) for \(s<j\), such that if, alongside the systems (7), one considers the systems
\[ x_{jk}^*=Q_{jk}^n(\mathbf x_{jk-1}^*,\mathbf F_j)+\delta_k^i\delta x_{ji}; \tag{7*} \]
\[ \delta_k^i=1\ \text{for } k=i;\qquad \delta_k^i=0\ \text{for } k\ne i;\qquad i=1,\ldots,m_j, \]
in which \(\delta x_{ji}\) in absolute value do not exceed some fi-

fixed positive number \(\varepsilon\), then the differences \(\delta x_{jk}(\delta x_{ji})\equiv x_{jk}^*-x_{jk}\) for all \(n\geqslant N\) satisfy the inequality

\[ |\delta x_{jk}(\delta x_{ji})|\leqslant C_2\varepsilon,\quad 1\leqslant i\leqslant k\leqslant m_j;\quad j=1,2,\ldots,l, \tag{8} \]

where the positive constant \(C_2\) depends neither on \(n\) nor on \(\varepsilon\).

We shall say that the c.a. \(Q\) is strongly stable on the r.d.e. (2) if it is weakly stable on this equation and, moreover, in the definition of weak stability the left-hand side of inequality (8) may be replaced by the quantity

\[ \Delta_{jk}=\sum_{i=1}^{k}|\delta x_{jk}(\delta x_{ji})|. \]

With every c.a. \(Q\) inverting the r.d.e. (2), one can associate a c.a. \(\overline Q\), consisting in the application of the c.a. \(Q\) to the r.d.e.

\[ l^n z^n=g^n, \tag{\(\overline{2}\)} \]

which is obtained from system (2) if in it one reverses the numbering of the equations and unknowns, i.e. if one introduces new numbers \(j\) for the equations and unknowns of system (2) by the formula \(j=n+1-i\), where \(i\) is the old number of an equation or unknown of system (2).

On one and the same r.d.e. one of the c.a. \(Q,\overline Q\) may be unstable, but the other stable. Proceeding from this, we shall say that the c.a. \(Q\) is weakly (strongly) stable in the class \(\mathfrak M(\omega_1,\omega_2)\times\mathfrak N(\omega_3)\), if on each r.d.e. from this class the conditions of weak (strong) stability are satisfied by at least one of the c.a. \(Q,\overline Q\). We shall say that the c.a. \(Q\) is strongly unstable in the class \(\mathfrak M(\omega_1,\omega_2)\times\mathfrak N(\omega_3)\), if for any number \(q\) there is an r.d.e. from this class for which there exist a positive number \(C\) and arbitrarily large indices \(n\), for which both in the case of the c.a. \(Q\) and in the case of the c.a. \(\overline Q\) the inequality

\[ \max_{i,j,k}|\delta x_{jk}(\delta x_{ji})|\geqslant Cn^q \]

is satisfied.

Consider the following computational algorithms. The formal application of the schemes of single division and of the compact scheme of the Gauss method (see \((^2)\)) to the solution of system (2) will be called, respectively, the c.a. \(B_1\) and the c.a. \(B_2\). Such formal use of universal methods cannot, of course, take into account the specifics of the structure of the matrices \(l^n\) and is accompanied by a considerable number of “parasitic” arithmetic operations, the results of which can be found without computation. Freeing the c.a. \(B_1\) and the c.a. \(B_2\) from such operations leads to one and the same c.a. \(B_3\)

\[ \sigma_i=-\frac{c_i^n}{b_i^n+a_i^n\sigma_{i-1}},\quad i=2,\ldots,n-1;\quad \sigma_1=-\frac{c_1^n}{b_1^n}; \]

\[ \psi_i=-\frac{f_i^n-a_i^n\psi_{i-1}}{b_i^n+a_i^n\sigma_{i-1}},\quad i=2,\ldots,n;\quad \psi_1=-\frac{f_1^n}{b_1^n}; \]

\[ y_i^n=\psi_i-\sigma_i y_{i+1}^n,\quad i=1,\ldots,n-1;\quad y_n^n=\psi_n. \]

An identical c.a. can also be obtained by starting from the method of difference factorization \((^3)\).

It can be shown that the exact values of the quantities \(\sigma_i\) in the computational algorithms \(B_1,B_2,B_3\) are negative and do not exceed \(1\) in absolute value. We shall assume that the actually obtained values \(\sigma_i^*\) are not less than \(-1\). This can always be achieved by introducing into the c.a. the operation of taking the maximum of the numbers \(-1\) and \(\sigma_i\). Then the following theorems are valid; in their formulations, by the c.a. \(B\) one may understand any of the c.a. \(B_1,B_2,B_3\).

Theorem 3. Let \(\omega_1=0,\ \omega_2=0\). The computational algorithm \(B\) is strongly unstable in the class \(\mathfrak M(\omega_1,\omega_2)\times \mathfrak M(\omega_3)\), whatever \(\omega_3\) may be.

Theorem 4. Let \(0<\omega_1\leqslant 1,\ \omega_2=0\). In order that the computational algorithm \(B\) be weakly stable in the class \(\mathfrak M(\omega_1,\omega_2)\times \mathfrak M(\omega_3)\), it is necessary and sufficient that \(\omega_3\leqslant -2\).

Theorem 5. Let \(\omega_1>1,\ \omega_2=0\). In order that the computational algorithm \(B\) be weakly stable in the class \(\mathfrak M(\omega_1,\omega_2)\times \mathfrak M(\omega_3)\), it is necessary and sufficient that \(\omega_3\leqslant -1\).

Theorem 6. Let \(\omega_1>0,\ \omega_2>0,\ \omega_3\leqslant 0\). The computational algorithm \(B\) is strongly stable in the class \(\mathfrak M(\omega_1,\omega_2)\times \mathfrak M(\omega_3)\).

In the proof of Theorems 4, 5, and 6, Theorem 2 is used.

Consider on the interval \(0\leqslant x\leqslant 1\) the boundary-value problem

\[ Mu\equiv (Au_x)_x+Bu_x+Cu=D,\qquad u(0)=u(1)=0. \]

Let the coefficients \(A,B,C,D\) and the solution \(u\) be sufficiently smooth functions of the variable \(x\), with \(A(x)\geqslant \gamma>0,\ C(x)\leqslant 0\). Let \(\Delta x=\dfrac{1}{n+1},\ v_i\equiv v\!\left(\dfrac{i}{n+1}\right)\). Replace the original problem by the problem of solving the finite-difference equation

\[ l_3^n y^n=f^n, \tag{9} \]

defined by the equalities

\[ a_i^n=A_{i-1}+4A_i-A_{i+1}-2\Delta x B_i,\qquad b_i^n=8A_i+4(\Delta x)^2C_i, \]

\[ c_i^n=-A_{i-1}+4A_i+A_{i+1}+2\Delta x B_i; \tag{10} \]

\[ f_i^n=4(\Delta x)^2D_i. \tag{11} \]

Theorem 7. The finite-difference equation (8) belongs to the class \(\mathfrak M(\omega_1,0)\times \mathfrak M(-2)\), \(0<\omega_1\leqslant 1\).

From Theorems 7 and 4 it follows that the computational algorithms \(B_1,\ B_2\), and \(B_3\) are weakly stable on the finite-difference equation (8).

We now consider the boundary-value problem for the parabolic equation in the domain \([0,1]\times[0,1]\)

\[ u_t-(Au_x)_x-Bu_x-Cu=D,\qquad u(0,t)=u(1,t)=0,\qquad u(x,0)=0. \]

We shall solve this problem by the method of grids, using the implicit scheme

\[ \frac{u_i^{j+1}-u_i^j}{\Delta t} = \frac{a_{ij+1}u_{i-1}^{j+1}-b_{ij+1}u_i^{j+1}+c_{ij+1}u_{i+1}^{j+1}}{4(\Delta x)^2} + D_i^{j+1}, \]

where \(D_i^j=D(i\Delta x,j\Delta t)\), and the coefficients \(a_{ij+1}, b_{ij+1}, c_{ij+1}\) are determined by the right-hand sides in formulas (10), which this time depend on the number of the time layer \(j+1\). Let \(\Delta x\equiv 1/(n+1),\ \Delta t=4r(\Delta x)^2,\ r>0\). Considering \(n\) as variable, we obtain, for all \(j\), the finite-difference equation

\[ l_n^n y^n=f^n. \tag{12} \]

Theorem 8. For all \(j\), the finite-difference equation (12) belongs to the class \(\mathfrak M(\omega_1,1)\times \mathfrak M(0)\), \(0<\omega_1\leqslant 1\).

From Theorems 8 and 6 it follows that the computational algorithms \(B_1,\ B_2\), and \(B_3\) are strongly stable on the finite-difference equation (12).

The author expresses sincere gratitude to O. A. Ladyzhenskaya for supervision of the work.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
20 VI 1962

REFERENCES

1. N. S. Bakhvalov, Computational Methods and Programming, 1, Moscow State University Press, 1962, p. 47.
2. D. K. Fadeev, V. N. Fadeeva, Computational Methods of Linear Algebra, 1960.
3. V. K. Saul’ev, Integration of Equations of Parabolic Type by the Method of Grids, 1960.