Relative conditioning of linear systems of ODEs with respect to perturbation in the matrix of the system and in the initial value

The thesis is about how perturbations in the initial value $y_{0}$ or in the coefficient matrix $A$ propagate along the solutions of $n$-dimensional linear ordinary differential equations (ODE) \begin{equation*} \left\{ \begin{array}{l} y^\prime(t) =Ay(t),\ t\geq 0,\\ y(0)=y_0, \end{array} \right. \end{equation*} where $A \in \mathbb{R}^{n\times n} $ and $y_0\in \mathbb{R}^n$ and $y(t)=e^{tA}y_0$ is the solution of the equation.\\ We begin by considering a perturbation analysis when the initial value $y_0$ is perturbed to $\tilde{y}_0$ with relative error \[ \varepsilon=\frac{\norm{\tilde{y}_0-y_0}}{\norm{y_0}}, \] where $\norm{\cdot}$ is a vector norm on $\mathbb{R}^n$. Due to perturbation in the initial value, the solution $y(t)=e^{tA}y_0$ is perturbed to $\tilde{y}(t)=e^{tA}\tilde{y}_0$ with relative error \[ \delta(t)=\frac{\left\| e^{tA}\tilde{y}_0-e^{tA}y_0\right\|}{\left\| e^{tA}y_0\right\|}. \] In other words, it is the (relative) conditioning of the problem \begin{equation*} y_0\mapsto e^{tA}y_0. \end{equation*} The relation between the error $\varepsilon$ and the error $\delta(t)$ is described by three condition numbers namely: the condition number with the direction of perturbation, the condition number independent of the direction of perturbation and the condition number not only independent of the specific direction of perturbation but also independent of the specific initial value. How these condition numbers behave over a long period of time is an important aspect of the study. In the thesis, we move towards perturbations in the matrix as well as componentwise relative errors, rather than normwise relative errors, for perturbations of the initial value. About the first topic of the thesis, we look over how perturbations propagate along the solution of the ODE, when it is the coefficient matrix $A$ rather than the initial value that perturbs. In other words, the interest is to study the conditioning of the problem $$ A\mapsto e^{tA}y_0. $$ In case when the matrix $A$ perturbs to $\tilde{A}$, the relative error is given by \[ \epsilon=\frac{\vertiii{\tilde{A}-A}}{\vertiii{A}} \] and the relative error in the solution of the ODE is given by \[ \xi(t)=\frac{\left\| e^{t\widetilde{A}}y_0-e^{tA}y_0\right\|}{\left\| e^{tA}y_0\right\|}. \] We introduce three condition numbers as before. The analysis of the condition numbers is done for a normal matrix $A$ and by making use of $2$-norm. We give very useful upper and lower bounds on these three condition numbers and we study their asymptotic behavior as time goes to infinity. There could be cases when someone is interested in the relative errors \[ \delta_l(t)=\frac{\vert \tilde{y}_l(t)-y_l(t) \vert}{\vert y_l(t) \vert}, \quad l=1,\dots,n, \] of the perturbed solution components. With the motivation that componentwise relative errors give more information than the normwise relative error, we make a componentwise relative error analysis, which is the other topic of this thesis. We consider perturbations in initial value $y_0$ with normwise relative error $\varepsilon$ and the relative error in the components of the solution of the equation given by $\delta_l(t)$. The interest is to study, for the $l$-th component, the conditioning of the problem $$ y_0\mapsto y_l(t)=e_l^Te^{tA}y_0, $$ where $e_l^T$ is the $l$-th vector of the canonical basis of $\mathbb{R}^n$. We make this analysis for a diagonalizable matrix $A$, diagonalizability being a generic situation for the matrix $A$. We give two condition numbers in this part of the thesis and study their asymptotic behavior as time goes to infinity.

The thesis is about how perturbations in the initial value $y_{0}$ or in the coefficient matrix $A$ propagate along the solutions of $n$-dimensional linear ordinary differential equations (ODE) \begin{equation*} \left\{ \begin{array}{l} y^\prime(t) =Ay(t),\ t\geq 0,\\ y(0)=y_0, \end{array} \right. \end{equation*} where $A \in \mathbb{R}^{n\times n} $ and $y_0\in \mathbb{R}^n$ and $y(t)=e^{tA}y_0$ is the solution of the equation.\\ We begin by considering a perturbation analysis when the initial value $y_0$ is perturbed to $\tilde{y}_0$ with relative error \[ \varepsilon=\frac{\norm{\tilde{y}_0-y_0}}{\norm{y_0}}, \] where $\norm{\cdot}$ is a vector norm on $\mathbb{R}^n$. Due to perturbation in the initial value, the solution $y(t)=e^{tA}y_0$ is perturbed to $\tilde{y}(t)=e^{tA}\tilde{y}_0$ with relative error \[ \delta(t)=\frac{\left\| e^{tA}\tilde{y}_0-e^{tA}y_0\right\|}{\left\| e^{tA}y_0\right\|}. \] In other words, it is the (relative) conditioning of the problem \begin{equation*} y_0\mapsto e^{tA}y_0. \end{equation*} The relation between the error $\varepsilon$ and the error $\delta(t)$ is described by three condition numbers namely: the condition number with the direction of perturbation, the condition number independent of the direction of perturbation and the condition number not only independent of the specific direction of perturbation but also independent of the specific initial value. How these condition numbers behave over a long period of time is an important aspect of the study. In the thesis, we move towards perturbations in the matrix as well as componentwise relative errors, rather than normwise relative errors, for perturbations of the initial value. About the first topic of the thesis, we look over how perturbations propagate along the solution of the ODE, when it is the coefficient matrix $A$ rather than the initial value that perturbs. In other words, the interest is to study the conditioning of the problem $$ A\mapsto e^{tA}y_0. $$ In case when the matrix $A$ perturbs to $\tilde{A}$, the relative error is given by \[ \epsilon=\frac{\vertiii{\tilde{A}-A}}{\vertiii{A}} \] and the relative error in the solution of the ODE is given by \[ \xi(t)=\frac{\left\| e^{t\widetilde{A}}y_0-e^{tA}y_0\right\|}{\left\| e^{tA}y_0\right\|}. \] We introduce three condition numbers as before. The analysis of the condition numbers is done for a normal matrix $A$ and by making use of $2$-norm. We give very useful upper and lower bounds on these three condition numbers and we study their asymptotic behavior as time goes to infinity. There could be cases when someone is interested in the relative errors \[ \delta_l(t)=\frac{\vert \tilde{y}_l(t)-y_l(t) \vert}{\vert y_l(t) \vert}, \quad l=1,\dots,n, \] of the perturbed solution components. With the motivation that componentwise relative errors give more information than the normwise relative error, we make a componentwise relative error analysis, which is the other topic of this thesis. We consider perturbations in initial value $y_0$ with normwise relative error $\varepsilon$ and the relative error in the components of the solution of the equation given by $\delta_l(t)$. The interest is to study, for the $l$-th component, the conditioning of the problem $$ y_0\mapsto y_l(t)=e_l^Te^{tA}y_0, $$ where $e_l^T$ is the $l$-th vector of the canonical basis of $\mathbb{R}^n$. We make this analysis for a diagonalizable matrix $A$, diagonalizability being a generic situation for the matrix $A$. We give two condition numbers in this part of the thesis and study their asymptotic behavior as time goes to infinity.