# Changing a Wide Matrix from Landscape to Portrait Orientation

\documentclass[a4paper,12pt,]{book}
\usepackage{graphicx}
\usepackage[inner=1.3cm, outer=1.3cm, top=2cm, bottom=2cm, bindingoffset=1cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathtools}
\begin{document}
\title{Chapter 3}
\author{Akor Eleojo Rachel}
\maketitle
Consider the Linear Multi-step Method:
$$\begin{split} y(x) =& \alpha_{0}(x)y_{n} + h\bigg[\beta_{0}(x)f_{n}+\beta_{1}(x)f_{n+1}+\beta_{\frac{3}{2}}(x)f_{n+\frac{3}{2}}+\beta_{2}(x)f_{n+2}+\beta_{\frac{5}{2}}(x)f_{n+\frac{5}{2}}+\beta_{3}(x)f_{n+3}+\beta_{\frac{7}{2}}(x)f_{n+\frac{7}{2}} \\ =\ &+\beta_{4}(x)f_{n+4}+\beta_{\frac{9}{2}}(x)f_{n+\frac{9}{2}}\bigg] \end{split}$$
$$A=\begin{bmatrix} 1 & {x_n} & {{x_n}^{2}} & {{x_n}^{3}} & {{x_n}^{4}} & {{x_n}^{5}} & {{x_n}^{6}} & {{x_n}^{7}} & {{x_n}^{8}} & {{x_n}^{9}}\\ 0 & 1 & 2 {x_n} & 3 {{x_n}^{2}} & 4 {{x_n}^{3}} & 5 {{x_n}^{4}} & 6 {{x_n}^{5}} & 7 {{x_n}^{6}} & 8 {{x_n}^{7}} & 9 {{x_n}^{8}}\\ 0 & 1 & 2 {x_{n+1}} & 3 {{x_{n+1}}^{2}} & 4 {{x_{n+1}}^{3}} & 5 {{x_{n+1}}^{4}} & 6{{x_{n+1}}^{5}} & 7 {{x_{n+1}}^{6}} & 8 {{x_{ n+1}}^{7}} & 9 {{x_{n+1}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{3}{2}}} & 3 {{x_{n+\frac{3}{2}}}^{2}}& 4 {{x_{n+\frac{3}{2}}}^{3}} & 5 {{x_{n+\frac{3}{2}}}^{4}} & 6 {{x_{n+\frac{3}{2}}}^{5}} & 7 {{x_{n+\frac{3}{2}}}^{6}} & 8 {{x_{n+\frac{3}{2}}}^{7}} & 9 {{x_{n+\frac{3}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+2}}& 3 {{x_{n+2}}^{2}} & 4 {{x_{n+2}}^{3}} & 5 {{x_{n+2}}^{4}} & 6 {{x_{ n+2}}^{5}} & 7 {{x_{n+2}}^{6}} & 8 {{x_{n+2}}^{7}} & 9 {{x_{n+2}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{5}{2}}} & 3 {{x_{n+\frac{5}{2}}}^{2}} & 4 {{x_{n+\frac{5}{2}}}^{3}} & 5{{x_{n+\frac{5}{2}}}^{4}} & 6 {{x_{n+\frac{5}{2}}}^{5}} & 7 {{x_{n+\frac{5}{2}}}^{6}} & 8 {{x_{n+\frac{5}{2}}}^{7}} & 9 {{x_{n+\frac{5}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+3}} & 3 {{x_{n+3}}^{2}} & 4 {{x_{n+3}}^{3}} & 5 {{x_{n+3}}^{4}} & 6 {{x_{ n+3} }^{5}} & 7 {{x_{n+3}}^{6}} & 8 {{x_{n+3}}^{7}} & 9 {{x_{n+3}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{7}{2}}} & 3 {{x_{n+\frac{7}{2}}}^{2}} & 4 {{x_{n+\frac{7}{2}}}^{3}} & 5 {{x_{n+\frac{7}{2}}}^{4}} & 6 {{x_{n+\frac{7}{2}}}^{5}} & 7 {{x_{n+\frac{7}{2}}}^{6}} & 8 {{x_{n+\frac{7}{2}}}^{7}} & 9 {{x_{n+\frac{7}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+4}}& 3 {{x_{n+4}}^{2}} & 4 {{x_{n+4}}^{3}} & 5 {{x_{n+4}}^{4}} & 6{{x_{ n+4} }^{5}} & 7 {{x_{n+4}}^{6}} & 8 {{x_{n+4}}^{7}} & 9 {{x_{n+4}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{9}{2}}} & 3 {{x_{n+\frac{9}{2}}}^{2}} & 4 {{x_{n+\frac{9}{2}}}^{3}} & 5 {{x_{ n+\frac{9}{2}}}^{4}} & 6 {{x_{n+\frac{9}{2}}}^{5}} & 7 {{x_{n+\frac{9}{2}}}^{6}} & 8 {{x_{n+\frac{9}{2}}}^{7}} & 9 {{x_{n+\frac{9}{2}}}^{8}} \end{bmatrix}$$
Replacing: {\large$x_{n}, x_{n+\frac{3}{2}}, x_{n+2}, x_{n+\frac{5}{2}}, x_{n+3}, x_{n+\frac{7}{2}}, x_{n+4}, x_{n+\frac{9}{2}}$}\\
with: $x_{n+1}-h, x_{n+1}+\frac{1}{2}h, x_{n+1}+h, x_{n+1}+\frac{3}{2}h, x_{n+1}+2h, x_{n+1}+\frac{5}{2}h, x_{n+1}+3h, x_{n+1}+\frac{7}{2}h$\\
we obtain:
$$\label{mat} B=\begin{bmatrix} 1 & {x_{n+1}-h} & {{(x_{n+1}-h)}^{2}} & {{(x_{n+1}-h)}^{3}} & {{(x_{n+1}-h)}^{4}} & {{(x_{n+1}-h)}^{5}} & {{(x_{n+1}-h)}^{6}} & {{(x_{n+1}-h)}^{7}} & {{(x_{n+1}-h)}^{8}} & {{(x_{n+1}-h)}^{9}}\\ 0 & 1 & 2 {(x_{n+1}-h)} & 3 {{(x_{n+1}-h)}^{2}} & 4 {{(x_{n+1}-h)}^{3}} & 5 {{(x_{n+1}-h)}^{4}} & 6 {{(x_{n+1}-h)}^{5}} & 7 {{(x_{n+1}-h)}^{6}} & 8 {{(x_{n+1}-h)}^{7}} & 9 {{(x_{n+1}-h)}^{8}}\\ 0 & 1 & 2 {x_{n+1}} & 3 {{x_{n+1}}^{2}} & 4 {{x_{n+1}}^{3}} & 5 {{x_{n+1}}^{4}} & 6{{x_{n+1}}^{5}} & 7 {{x_{n+1}}^{6}} & 8 {{x_{ n+1}}^{7}} & 9 {{x_{n+1}}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{1}{2}h)} & 3 {{(x_{n+1}+\frac{1}{2}h)}^{2}}& 4 {{(x_{n+1}+\frac{1}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{1}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{1}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{1}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{1}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{1}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+h)}& 3 {{(x_{n+1}+h)}^{2}} & 4 {{(x_{n+1}+h)}^{3}} & 5 {{(x_{n+1}+h)}^{4}} & 6 {{(x_{n+1}+h)}^{5}} & 7 {{(x_{n+1}+h)}^{6}} & 8 {{(x_{n+1}+h)}^{7}} & 9 {{(x_{n+1}+h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{3}{2}h)} & 3 {{(x_{n+1}+\frac{3}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{3}{2}h)}^{3}} & 5{{(x_{n+1}+\frac{3}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{3}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{3}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{3}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{3}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+2h)} & 3 {{(x_{n+1}+2h)}^{2}} & 4 {{(x_{n+1}+2h)}^{3}} & 5 {{(x_{n+1}+2h)}^{4}} & 6 {{(x_{n+1}+2h)}^{5}} & 7 {{(x_{n+1}+2h)}^{6}} & 8 {{(x_{n+1}+2h)}^{7}} & 9 {{(x_{n+1}+2h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{5}{2}h)} & 3 {{(x_{n+1}+\frac{5}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{5}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{5}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{5}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{5}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{5}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{5}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+3h)}& 3 {{(x_{n+1}+3h)}^{2}} & 4 {{(x_{n+1}+3h)}^{3}} & 5 {{(x_{n+1}+3h)}^{4}} & 6{{(x_{n+1}+3h)}^{5}} & 7 {{(x_{n+1}+3h)}^{6}} & 8 {{(x_{n+1}+3h)}^{7}} & 9 {{(x_{n+1}+3h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{7}{2}h)} & 3 {{(x_{n+1}+\frac{7}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{7}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{7}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{7}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{7}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{7}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{7}{2}h)}^{8}} \end{bmatrix}$$
The new Matrix is renamed B for ease of reference.\\
\par
The determinant of B is:
$det (B) =\frac{15380234690625 {{h}^{36}}}{64}$
The inverse of (\ref{mat}) is the C matrix calculated using \textbf{Maple} codes as shown in the \textbf{Appendix}.
Our only interest in the C matrix is it's first row and the elements are:
\begin{align*}
&c_{11}=1\\
&c_{12}=-\frac{160 {x_{n+1}^9}+2520 h {x_{n+1}^8}+16560 {h^2} {x_{n+1}^7}+58800 {h^3} {x_{n+1}^6}+121842 {h^4} {x_{n+1}^5}+147735 {h^5} {x_{n+1}^4}+98010 {h^6} {x_{n+1sing}^3}+28350 {h^7} {x_{n+1}^2}-473977 {h^9}}{2041200 {h^8}}\\
&c_{13}=\frac{320 {x_{n+1}^9}+4680 h {x_{n+1}^8}+27360 {h^2} {x_{n+1}^7}+78960 {h^3} {x_{n+1}^6}+102564 {h^4} {x_{n+1}^5}-9135 {h^5} {x_{n+1}^4}-197940 {h^6} {x_{n+1}^3}-237330 {h^7} {x_{n+1}^2}-113400 {h^8} {x_{n+1}}+343921 {h^9}}{113400 {h^8}}\\
&c_{14}=-\frac{1120 {x_{n+1}^9}+15750 h {x_{n+1}^8}+86760 {h^2} {x_{n+1}^7}+225750 {h^3} {x_{n+1}^6}+223524 {h^4} {x_{n+1}^5}-171675 {h^5} {x_{n+1}^4}-578340 {h^6} {x_{n+1}^3}-396900 {h^7} {x_{n+1}^2}+594011 {h^9}}{85050 {h^8}}\\
&c_{15}=\frac{560 {x_{n+1}^{9}}+7560 h {x_{n+1}^{8}}+39240 {{h}^{2}} {x_{n+1}^{7}}+92400 {{h}^{3}} {x_{n+1}^{6}}+68607 {{h}^{4}} {x_{n+1}^{5}}-101745 {{h}^{5}} {x_{ n+1}^{4}}-210735 {{h}^{6}} {x_{n+1}^{3}}-99225 {{h}^{7}} {x_{n+1}^{2}}+203338 {{h}^{9}}}{18900 {{h}^{8}}} \\
&c_{16}=-\frac{1120 {x_{n+1}^{9}}+14490 h {x_{n+1}^{8}}+70920 {{h}^{2}} {x_{n+1}^{7}}+152250 {{h}^{3}} {x_{n+1}^{6}}+84924 {{h}^{4}} {x_{n+1}^{5}}-191205 {{h}^{5}} {x_{n+1}^{4}}-310380 {{h}^{6}} {x_{n+1}^{3}}-132300 {{h}^{7}} {x_{n+1}^{2}}+310181 {{h}^{9}}}{28350 {{h}^{8}}}\\
&c_{17}=\frac{2240 {x_{n+1}^{9}}+27720 h {x_{n+1}^{8}}+128160 {{h}^{2}} {x_{n+1}^{7}}+253680 {{h}^{3}} {x_{n+1}^{6}}+109116 {{h}^{4}} {x_{n+1}^{5}}-336735 {{h}^{5}} {x_{n+1}^{4}}-487620 {{h}^{6}} {x_{n+1}^{3}}-198450 {{h}^{7}} {x_{n+1}^{2}}+501889 {{h}^{9}}}{68040 {{h}^{8}}}\\
&c_{18}=-\frac{160 {x_{n+1}^{9}}+1890 h {x_{n+1}^{8}}+8280 {{h}^{2}} {x_{n+1}^{7}}+15330 {{h}^{3}} {x_{n+1}^{6}}+5292 {{h}^{4}} {x_{n+1}^{5}}-21105 {{h}^{5}} {x_{n+1}^{4}}-28620 {{h}^{6}} {x_{n+1}^3}-11340 {{h}^{7}} {x_{n+1}^{2}}+30113 {{h}^{9}}}{9450 {h^8}}\\
&c_{19}=\frac{1120 {x_{n+1}^9}+12600 h {x_{n+1}^8}+52560 {h^2} {x_{n+1}^7}+92400 {h^3} {x_{n+1}^6}+26334 {h^4} {x_{n+1}^5}-130725 {h^5} {x_{n+1}^4}-169890 {h^6} {x_{n+1}^3}-66150 {h^7} {x_{n+1}^2}+181751 {h^9}}{226800 {h^8}}\\
&c_{(10)}=-\frac{160 {x_{n+1}^9}+1710 h {x_{n+1}^8}+6840 {h^2} {x_{n+1}^7}+11550 {h^3} {x_{n+1}^6}+2772 {h^4} {x_{n+1}^5}-16695 {h^5} {x_{n+1}^4}-21060 {h^6} {x_{n+1}^3}-8100 {h^7}{x_{n+1}^2}+22823 {h^9}}{255150{h^8}}
\end{align*}
From the above, we obtain the following continuous coefficients-This is obtained through Maple Codes as shown in the \textbf{Appendix}:
\begin{align*}
&\alpha_{0}(x)=y_{n}\\
&h\beta_{0}(x)=\frac{-160 {{w}^{9}}-2520 h {{w}^{8}}-16560 {{h}^{2}} {{w}^{7}}-58800 {{h}^{3}} {{w}^{6}}-121842 {{h}^{4}} {{w}^{5}}-147735 {{h}^{5}} {{w}^{4}}-98010 {{h}^{6}} {{w}^{3}}-28350 {{h}^{7}} {{w}^{2}}+473977 {{h}^{9}}}{2041200{h^8}}\\
&h\beta_{1}(x)=\frac{320 {{w}^{9}}+4680 h {{w}^{8}}+27360 {{h}^{2}} {{w}^{7}}+78960 {{h}^{3}} {{w}^{6}}+102564 {{h}^{4}} {{w}^{5}}-9135 {{h}^{5}} {{w}^{4}}-197940 {{h}^{6}} {{w}^{3}}-237330 {{h}^{7}} {{w}^{2}}-113400 {{h}^{8}} w+343921 {{h}^{9}}}{113400{h^8}}\\
&h\beta_{\frac{3}{2}}(x)=\frac{-1120 {{w}^{9}}-15750 h {{w}^{8}}-86760 {{h}^{2}} {{w}^{7}}-225750 {{h}^{3}} {{w}^{6}}-223524 {{h}^{4}} {{w}^{5}}+171675 {{h}^{5}} {{w}^{4}}+578340 {{h}^{6}} {{w}^{3}}+396900 {{h}^{7}} {{w}^{2}}-594011 {{h}^{9}}}{85050{h^8}}\\
&h\beta_{2}(x)=\frac{560 {{w}^{9}}+7560 h {{w}^{8}}+39240 {{h}^{2}} {{w}^{7}}+92400 {{h}^{3}} {{w}^{6}}+68607 {{h}^{4}} {{w}^{5}}-101745 {{h}^{5}} {{w}^{4}}-210735 {{h}^{6}} {{w}^{3}}-99225 {{h}^{7}} {{w}^{2}}+203338 {{h}^{9}}}{18900{h^8}}\\
&h\beta_{\frac{5}{2}}(x)=\frac{-1120 {w^9}-14490 h {w^8}-70920 {h^2} {w^7}-152250 {h^3} {w^6}-84924 {h^4} {w^5}+191205 {h^5} {w^4}+310380 {h^6} {w^3}+132300 {h^7} {w^2}-310181 {h^9}}{28350{h^8}}\\
&h\beta_{3}(x)=\frac{2240 {{w}^{9}}+27720 h {{w}^{8}}+128160 {{h}^{2}} {{w}^{7}}+253680 {{h}^{3}} {{w}^{6}}+109116 {{h}^{4}} {{w}^{5}}-336735 {{h}^{5}} {{w}^{4}}-487620 {{h}^{6}} {{w}^{3}}-198450 {{h}^{7}} {{w}^{2}}+501889 {{h}^{9}}}{68040{h^8}}\\
&h\beta_{\frac{7}{2}}(x)=\frac{-160 {{w}^{9}}-1890 h {{w}^{8}}-8280 {{h}^{2}} {{w}^{7}}-15330 {{h}^{3}} {{w}^{6}}-5292 {{h}^{4}} {{w}^{5}}+21105 {{h}^{5}} {{w}^{4}}+28620 {{h}^{6}} {{w}^{3}}+11340 {{h}^{7}} {{w}^{2}}-30113 {{h}^{9}}}{9450{h^8}}\\
&h\beta_{4}(x)=\frac{1120 {{w}^{9}}+12600 h {{w}^{8}}+52560 {{h}^{2}} {{w}^{7}}+92400 {{h}^{3}} {{w}^{6}}+26334 {{h}^{4}} {{w}^{5}}-130725 {{h}^{5}} {{w}^{4}}-169890 {{h}^{6}} {{w}^{3}}-66150 {{h}^{7}} {{w}^{2}}+181751 {{h}^{9}}}{226800{h^8}}\\
&h\beta_{\frac{9}{2}}(x)=\frac{-160 {{w}^{9}}-1710 h {{w}^{8}}-6840 {{h}^{2}} {{w}^{7}}-11550 {{h}^{3}} {{w}^{6}}-2772 {{h}^{4}} {{w}^{5}}+16695 {{h}^{5}} {{w}^{4}}+21060 {{h}^{6}} {{w}^{3}}+8100 {{h}^{7}} {{w}^{2}}-22823 {{h}^{9}}}{255150{h^8}}
\end{align*}
Evaluating the above at $w=0, w=-\frac{h}{2}, w=-h, w=-\frac{3h}{2}, w=-2, w=-\frac{5h}{2}, w=-3, w=-\frac{7h}{2}$ to obtain the following discrete schemes:
\begin{align*}
&y_{n+1}=\frac{y_{n}}{2041200}+\frac{h}{2041200}\bigg[473977{f_n} +6190578f_{n+1}-14256264  f_{n+\frac{3}{2}}+21960504f_{n+2}-22333032f_{n+\frac{5}{2}}+15056670f_{ n+3}\\
&-6504408 f_{ n+\frac{7}{2}} +1635759 f_{ n+4}-182584 f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{3}{2}}=\frac{y_{n}}{89600}+\frac{h}{89600}\bigg[20759 {f_n} +287046  f_{ n+1} -581818  f_{ n+\frac{3}{2}}+936468  f_{ n+2}-958194  f_{ n+\frac{5}{2}} +647690  f_{ n+3} -280206f_{n+\frac{7}{2}}  +70533  f_{ n+4}-7878  f_{ n+\frac{9}{2}}\bigg]\\
&y_{n+2}=\frac{y_{n}}{255150}+\frac{h}{255150}\bigg[59143{f_n}+814932f_{n+1}-1601616f_{\frac{n+3}{2}}+2762856f_{n+2}-2761488f_{\frac{n+5}{2}}+1860780f_{n+3}-803952f_{\frac{n+7}{2}}+202221f_{n+4}-22576f_{\frac{n+9}{2}}\bigg]\\
&y_{n+\frac{5}{2}}=\frac{y_{n}}{2612736}+\frac{h}{2612736}\bigg[605495{f_n}+8353350  f_{ n+1}-16467450  f_{ n+\frac{3}{2}}+28962900  f_{ n+2}-27460530-f_{ n+\frac{5}{2}}+18890250  f_{ n+3}8182350  f_{ n+\frac{7}{2}}\\
&+2060325f_{ n+4}-230150f_{ n+\frac{9}{2}}\bigg]\\
&y_{n+3}=\frac{y_{n}}{2800}+\frac{h}{2800}\bigg[649{f_n}+8946f_{n+1}-17608f_{n+\frac{3}{2}}+30888f_{n+2}-28584f_{n+\frac{5}{2}}+20990f_{n+3}-8856f_{n+\frac{7}{2}}+2223f_{n+4}-248f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{7}{2}}=\frac{y_{n}}{9331200}+\frac{h}{9331200}\bigg[2162377{f_n}+29837178f_{n+1}-58823814f_{n+\frac{3}{2}}+103389804f_{n+2}-96271182f_{n+\frac{5}{2}}+73295670f_{n+3}-27390258f_{n+\frac{7}{2}}\\&+7276059f_{n+4}-816634f_{n+\frac{9}{2}}\bigg]\\
&y_{n+4}=\frac{y_{n}}{127575}+\frac{h}{127575}\bigg[29578{f_n}+407232f_{n+1}-800256f_{n+\frac{3}{2}}+1402056f_{n+2}-1294848f_{n+\frac{5}{2}}+972480f_{n+3}-317952f_{n+\frac{7}{2}}+123786f_{n+4}-11776f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{9}{2}}=\frac{y_{n}}{89600}+\frac{h}{89600}\bigg[20727{f_n}+288198f_{n+1}-574074f_{n+\frac{3}{2}}+1017684f_{n+2}-965682f_{n+\frac{5}{2}}+748170f_{n+3}-278478f_{n+\frac{7}{2}}+141669f_{n+4}4986f_{n+\frac{9}{2}}\bigg]
\end{align*}
\end{document}

I initially did this work in landscape, but I need it in portrait. I am at wits end and in a time crunch. Below is the work I initially did using landscape mode

\documentclass[a4paper,12pt,landscape]{book}
\usepackage[inner=1.3cm, outer=1.3cm, top=2cm, bottom=2cm, bindingoffset=1cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathtools}
\begin{document}
\title{Chapter 3}
\author{Akor Eleojo Rachel}
\maketitle
Consider the Linear Multi-step Method:
$$\begin{split} y(x) =& \alpha_{0}(x)y_{n} + h\bigg[\beta_{0}(x)f_{n}+\beta_{1}(x)f_{n+1}+\beta_{\frac{3}{2}}(x)f_{n+\frac{3}{2}}+\beta_{2}(x)f_{n+2}+\beta_{\frac{5}{2}}(x)f_{n+\frac{5}{2}}+\beta_{3}(x)f_{n+3}+\beta_{\frac{7}{2}}(x)f_{n+\frac{7}{2}} \\ =\ &+\beta_{4}(x)f_{n+4}+\beta_{\frac{9}{2}}(x)f_{n+\frac{9}{2}}\bigg] \end{split}$$
$$A=\begin{bmatrix} 1 & {x_n} & {{x_n}^{2}} & {{x_n}^{3}} & {{x_n}^{4}} & {{x_n}^{5}} & {{x_n}^{6}} & {{x_n}^{7}} & {{x_n}^{8}} & {{x_n}^{9}}\\ 0 & 1 & 2 {x_n} & 3 {{x_n}^{2}} & 4 {{x_n}^{3}} & 5 {{x_n}^{4}} & 6 {{x_n}^{5}} & 7 {{x_n}^{6}} & 8 {{x_n}^{7}} & 9 {{x_n}^{8}}\\ 0 & 1 & 2 {x_{n+1}} & 3 {{x_{n+1}}^{2}} & 4 {{x_{n+1}}^{3}} & 5 {{x_{n+1}}^{4}} & 6{{x_{n+1}}^{5}} & 7 {{x_{n+1}}^{6}} & 8 {{x_{ n+1}}^{7}} & 9 {{x_{n+1}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{3}{2}}} & 3 {{x_{n+\frac{3}{2}}}^{2}}& 4 {{x_{n+\frac{3}{2}}}^{3}} & 5 {{x_{n+\frac{3}{2}}}^{4}} & 6 {{x_{n+\frac{3}{2}}}^{5}} & 7 {{x_{n+\frac{3}{2}}}^{6}} & 8 {{x_{n+\frac{3}{2}}}^{7}} & 9 {{x_{n+\frac{3}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+2}}& 3 {{x_{n+2}}^{2}} & 4 {{x_{n+2}}^{3}} & 5 {{x_{n+2}}^{4}} & 6 {{x_{ n+2}}^{5}} & 7 {{x_{n+2}}^{6}} & 8 {{x_{n+2}}^{7}} & 9 {{x_{n+2}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{5}{2}}} & 3 {{x_{n+\frac{5}{2}}}^{2}} & 4 {{x_{n+\frac{5}{2}}}^{3}} & 5{{x_{n+\frac{5}{2}}}^{4}} & 6 {{x_{n+\frac{5}{2}}}^{5}} & 7 {{x_{n+\frac{5}{2}}}^{6}} & 8 {{x_{n+\frac{5}{2}}}^{7}} & 9 {{x_{n+\frac{5}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+3}} & 3 {{x_{n+3}}^{2}} & 4 {{x_{n+3}}^{3}} & 5 {{x_{n+3}}^{4}} & 6 {{x_{ n+3} }^{5}} & 7 {{x_{n+3}}^{6}} & 8 {{x_{n+3}}^{7}} & 9 {{x_{n+3}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{7}{2}}} & 3 {{x_{n+\frac{7}{2}}}^{2}} & 4 {{x_{n+\frac{7}{2}}}^{3}} & 5 {{x_{n+\frac{7}{2}}}^{4}} & 6 {{x_{n+\frac{7}{2}}}^{5}} & 7 {{x_{n+\frac{7}{2}}}^{6}} & 8 {{x_{n+\frac{7}{2}}}^{7}} & 9 {{x_{n+\frac{7}{2}}}^{8}}\\ 0 & 1 & 2 {x_{n+4}}& 3 {{x_{n+4}}^{2}} & 4 {{x_{n+4}}^{3}} & 5 {{x_{n+4}}^{4}} & 6{{x_{ n+4} }^{5}} & 7 {{x_{n+4}}^{6}} & 8 {{x_{n+4}}^{7}} & 9 {{x_{n+4}}^{8}}\\ 0 & 1 & 2 {x_{n+\frac{9}{2}}} & 3 {{x_{n+\frac{9}{2}}}^{2}} & 4 {{x_{n+\frac{9}{2}}}^{3}} & 5 {{x_{ n+\frac{9}{2}}}^{4}} & 6 {{x_{n+\frac{9}{2}}}^{5}} & 7 {{x_{n+\frac{9}{2}}}^{6}} & 8 {{x_{n+\frac{9}{2}}}^{7}} & 9 {{x_{n+\frac{9}{2}}}^{8}} \end{bmatrix}$$
Replacing: {\large$x_{n}, x_{n+\frac{3}{2}}, x_{n+2}, x_{n+\frac{5}{2}}, x_{n+3}, x_{n+\frac{7}{2}}, x_{n+4}, x_{n+\frac{9}{2}}$}\\
with: $x_{n+1}-h, x_{n+1}+\frac{1}{2}h, x_{n+1}+h, x_{n+1}+\frac{3}{2}h, x_{n+1}+2h, x_{n+1}+\frac{5}{2}h, x_{n+1}+3h, x_{n+1}+\frac{7}{2}h$\\
we obtain:
$$\label{mat} B=\begin{bmatrix} 1 & {x_{n+1}-h} & {{(x_{n+1}-h)}^{2}} & {{(x_{n+1}-h)}^{3}} & {{(x_{n+1}-h)}^{4}} & {{(x_{n+1}-h)}^{5}} & {{(x_{n+1}-h)}^{6}} & {{(x_{n+1}-h)}^{7}} & {{(x_{n+1}-h)}^{8}} & {{(x_{n+1}-h)}^{9}}\\ 0 & 1 & 2 {(x_{n+1}-h)} & 3 {{(x_{n+1}-h)}^{2}} & 4 {{(x_{n+1}-h)}^{3}} & 5 {{(x_{n+1}-h)}^{4}} & 6 {{(x_{n+1}-h)}^{5}} & 7 {{(x_{n+1}-h)}^{6}} & 8 {{(x_{n+1}-h)}^{7}} & 9 {{(x_{n+1}-h)}^{8}}\\ 0 & 1 & 2 {x_{n+1}} & 3 {{x_{n+1}}^{2}} & 4 {{x_{n+1}}^{3}} & 5 {{x_{n+1}}^{4}} & 6{{x_{n+1}}^{5}} & 7 {{x_{n+1}}^{6}} & 8 {{x_{ n+1}}^{7}} & 9 {{x_{n+1}}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{1}{2}h)} & 3 {{(x_{n+1}+\frac{1}{2}h)}^{2}}& 4 {{(x_{n+1}+\frac{1}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{1}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{1}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{1}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{1}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{1}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+h)}& 3 {{(x_{n+1}+h)}^{2}} & 4 {{(x_{n+1}+h)}^{3}} & 5 {{(x_{n+1}+h)}^{4}} & 6 {{(x_{n+1}+h)}^{5}} & 7 {{(x_{n+1}+h)}^{6}} & 8 {{(x_{n+1}+h)}^{7}} & 9 {{(x_{n+1}+h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{3}{2}h)} & 3 {{(x_{n+1}+\frac{3}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{3}{2}h)}^{3}} & 5{{(x_{n+1}+\frac{3}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{3}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{3}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{3}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{3}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+2h)} & 3 {{(x_{n+1}+2h)}^{2}} & 4 {{(x_{n+1}+2h)}^{3}} & 5 {{(x_{n+1}+2h)}^{4}} & 6 {{(x_{n+1}+2h)}^{5}} & 7 {{(x_{n+1}+2h)}^{6}} & 8 {{(x_{n+1}+2h)}^{7}} & 9 {{(x_{n+1}+2h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{5}{2}h)} & 3 {{(x_{n+1}+\frac{5}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{5}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{5}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{5}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{5}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{5}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{5}{2}h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+3h)}& 3 {{(x_{n+1}+3h)}^{2}} & 4 {{(x_{n+1}+3h)}^{3}} & 5 {{(x_{n+1}+3h)}^{4}} & 6{{(x_{n+1}+3h)}^{5}} & 7 {{(x_{n+1}+3h)}^{6}} & 8 {{(x_{n+1}+3h)}^{7}} & 9 {{(x_{n+1}+3h)}^{8}}\\ 0 & 1 & 2 {(x_{n+1}+\frac{7}{2}h)} & 3 {{(x_{n+1}+\frac{7}{2}h)}^{2}} & 4 {{(x_{n+1}+\frac{7}{2}h)}^{3}} & 5 {{(x_{n+1}+\frac{7}{2}h)}^{4}} & 6 {{(x_{n+1}+\frac{7}{2}h)}^{5}} & 7 {{(x_{n+1}+\frac{7}{2}h)}^{6}} & 8 {{(x_{n+1}+\frac{7}{2}h)}^{7}} & 9 {{(x_{n+1}+\frac{7}{2}h)}^{8}} \end{bmatrix}$$
The new Matrix is renamed B for ease of reference.\\
\par
The determinant of B is:
$det (B) =\frac{15380234690625 {{h}^{36}}}{64}$
The inverse of (\ref{mat}) is the C matrix calculated using \textbf{Maple} codes as shown in the \textbf{Appendix}.
Our only interest in the C matrix is it's first row and the elements are:
\begin{align*}
&c_{11}=1\\
&c_{12}=-\frac{160 {x_{n+1}^9}+2520 h {x_{n+1}^8}+16560 {h^2} {x_{n+1}^7}+58800 {h^3} {x_{n+1}^6}+121842 {h^4} {x_{n+1}^5}+147735 {h^5} {x_{n+1}^4}+98010 {h^6} {x_{n+1sing}^3}+28350 {h^7} {x_{n+1}^2}-473977 {h^9}}{2041200 {h^8}}\\
&c_{13}=\frac{320 {x_{n+1}^9}+4680 h {x_{n+1}^8}+27360 {h^2} {x_{n+1}^7}+78960 {h^3} {x_{n+1}^6}+102564 {h^4} {x_{n+1}^5}-9135 {h^5} {x_{n+1}^4}-197940 {h^6} {x_{n+1}^3}-237330 {h^7} {x_{n+1}^2}-113400 {h^8} {x_{n+1}}+343921 {h^9}}{113400 {h^8}}\\
&c_{14}=-\frac{1120 {x_{n+1}^9}+15750 h {x_{n+1}^8}+86760 {h^2} {x_{n+1}^7}+225750 {h^3} {x_{n+1}^6}+223524 {h^4} {x_{n+1}^5}-171675 {h^5} {x_{n+1}^4}-578340 {h^6} {x_{n+1}^3}-396900 {h^7} {x_{n+1}^2}+594011 {h^9}}{85050 {h^8}}\\
&c_{15}=\frac{560 {x_{n+1}^{9}}+7560 h {x_{n+1}^{8}}+39240 {{h}^{2}} {x_{n+1}^{7}}+92400 {{h}^{3}} {x_{n+1}^{6}}+68607 {{h}^{4}} {x_{n+1}^{5}}-101745 {{h}^{5}} {x_{ n+1}^{4}}-210735 {{h}^{6}} {x_{n+1}^{3}}-99225 {{h}^{7}} {x_{n+1}^{2}}+203338 {{h}^{9}}}{18900 {{h}^{8}}} \\
&c_{16}=-\frac{1120 {x_{n+1}^{9}}+14490 h {x_{n+1}^{8}}+70920 {{h}^{2}} {x_{n+1}^{7}}+152250 {{h}^{3}} {x_{n+1}^{6}}+84924 {{h}^{4}} {x_{n+1}^{5}}-191205 {{h}^{5}} {x_{n+1}^{4}}-310380 {{h}^{6}} {x_{n+1}^{3}}-132300 {{h}^{7}} {x_{n+1}^{2}}+310181 {{h}^{9}}}{28350 {{h}^{8}}}\\
&c_{17}=\frac{2240 {x_{n+1}^{9}}+27720 h {x_{n+1}^{8}}+128160 {{h}^{2}} {x_{n+1}^{7}}+253680 {{h}^{3}} {x_{n+1}^{6}}+109116 {{h}^{4}} {x_{n+1}^{5}}-336735 {{h}^{5}} {x_{n+1}^{4}}-487620 {{h}^{6}} {x_{n+1}^{3}}-198450 {{h}^{7}} {x_{n+1}^{2}}+501889 {{h}^{9}}}{68040 {{h}^{8}}}\\
&c_{18}=-\frac{160 {x_{n+1}^{9}}+1890 h {x_{n+1}^{8}}+8280 {{h}^{2}} {x_{n+1}^{7}}+15330 {{h}^{3}} {x_{n+1}^{6}}+5292 {{h}^{4}} {x_{n+1}^{5}}-21105 {{h}^{5}} {x_{n+1}^{4}}-28620 {{h}^{6}} {x_{n+1}^3}-11340 {{h}^{7}} {x_{n+1}^{2}}+30113 {{h}^{9}}}{9450 {h^8}}\\
&c_{19}=\frac{1120 {x_{n+1}^9}+12600 h {x_{n+1}^8}+52560 {h^2} {x_{n+1}^7}+92400 {h^3} {x_{n+1}^6}+26334 {h^4} {x_{n+1}^5}-130725 {h^5} {x_{n+1}^4}-169890 {h^6} {x_{n+1}^3}-66150 {h^7} {x_{n+1}^2}+181751 {h^9}}{226800 {h^8}}\\
&c_{(10)}=-\frac{160 {x_{n+1}^9}+1710 h {x_{n+1}^8}+6840 {h^2} {x_{n+1}^7}+11550 {h^3} {x_{n+1}^6}+2772 {h^4} {x_{n+1}^5}-16695 {h^5} {x_{n+1}^4}-21060 {h^6} {x_{n+1}^3}-8100 {h^7}{x_{n+1}^2}+22823 {h^9}}{255150{h^8}}
\end{align*}
From the above, we obtain the following continuous coefficients-This is obtained through Maple Codes as shown in the \textbf{Appendix}:
\begin{align*}
&\alpha_{0}(x)=y_{n}\\
&h\beta_{0}(x)=\frac{-160 {{w}^{9}}-2520 h {{w}^{8}}-16560 {{h}^{2}} {{w}^{7}}-58800 {{h}^{3}} {{w}^{6}}-121842 {{h}^{4}} {{w}^{5}}-147735 {{h}^{5}} {{w}^{4}}-98010 {{h}^{6}} {{w}^{3}}-28350 {{h}^{7}} {{w}^{2}}+473977 {{h}^{9}}}{2041200{h^8}}\\
&h\beta_{1}(x)=\frac{320 {{w}^{9}}+4680 h {{w}^{8}}+27360 {{h}^{2}} {{w}^{7}}+78960 {{h}^{3}} {{w}^{6}}+102564 {{h}^{4}} {{w}^{5}}-9135 {{h}^{5}} {{w}^{4}}-197940 {{h}^{6}} {{w}^{3}}-237330 {{h}^{7}} {{w}^{2}}-113400 {{h}^{8}} w+343921 {{h}^{9}}}{113400{h^8}}\\
&h\beta_{\frac{3}{2}}(x)=\frac{-1120 {{w}^{9}}-15750 h {{w}^{8}}-86760 {{h}^{2}} {{w}^{7}}-225750 {{h}^{3}} {{w}^{6}}-223524 {{h}^{4}} {{w}^{5}}+171675 {{h}^{5}} {{w}^{4}}+578340 {{h}^{6}} {{w}^{3}}+396900 {{h}^{7}} {{w}^{2}}-594011 {{h}^{9}}}{85050{h^8}}\\
&h\beta_{2}(x)=\frac{560 {{w}^{9}}+7560 h {{w}^{8}}+39240 {{h}^{2}} {{w}^{7}}+92400 {{h}^{3}} {{w}^{6}}+68607 {{h}^{4}} {{w}^{5}}-101745 {{h}^{5}} {{w}^{4}}-210735 {{h}^{6}} {{w}^{3}}-99225 {{h}^{7}} {{w}^{2}}+203338 {{h}^{9}}}{18900{h^8}}\\
&h\beta_{\frac{5}{2}}(x)=\frac{-1120 {w^9}-14490 h {w^8}-70920 {h^2} {w^7}-152250 {h^3} {w^6}-84924 {h^4} {w^5}+191205 {h^5} {w^4}+310380 {h^6} {w^3}+132300 {h^7} {w^2}-310181 {h^9}}{28350{h^8}}\\
&h\beta_{3}(x)=\frac{2240 {{w}^{9}}+27720 h {{w}^{8}}+128160 {{h}^{2}} {{w}^{7}}+253680 {{h}^{3}} {{w}^{6}}+109116 {{h}^{4}} {{w}^{5}}-336735 {{h}^{5}} {{w}^{4}}-487620 {{h}^{6}} {{w}^{3}}-198450 {{h}^{7}} {{w}^{2}}+501889 {{h}^{9}}}{68040{h^8}}\\
&h\beta_{\frac{7}{2}}(x)=\frac{-160 {{w}^{9}}-1890 h {{w}^{8}}-8280 {{h}^{2}} {{w}^{7}}-15330 {{h}^{3}} {{w}^{6}}-5292 {{h}^{4}} {{w}^{5}}+21105 {{h}^{5}} {{w}^{4}}+28620 {{h}^{6}} {{w}^{3}}+11340 {{h}^{7}} {{w}^{2}}-30113 {{h}^{9}}}{9450{h^8}}\\
&h\beta_{4}(x)=\frac{1120 {{w}^{9}}+12600 h {{w}^{8}}+52560 {{h}^{2}} {{w}^{7}}+92400 {{h}^{3}} {{w}^{6}}+26334 {{h}^{4}} {{w}^{5}}-130725 {{h}^{5}} {{w}^{4}}-169890 {{h}^{6}} {{w}^{3}}-66150 {{h}^{7}} {{w}^{2}}+181751 {{h}^{9}}}{226800{h^8}}\\
&h\beta_{\frac{9}{2}}(x)=\frac{-160 {{w}^{9}}-1710 h {{w}^{8}}-6840 {{h}^{2}} {{w}^{7}}-11550 {{h}^{3}} {{w}^{6}}-2772 {{h}^{4}} {{w}^{5}}+16695 {{h}^{5}} {{w}^{4}}+21060 {{h}^{6}} {{w}^{3}}+8100 {{h}^{7}} {{w}^{2}}-22823 {{h}^{9}}}{255150{h^8}}
\end{align*}
Evaluating the above at $w=0, w=-\frac{h}{2}, w=-h, w=-\frac{3h}{2}, w=-2, w=-\frac{5h}{2}, w=-3, w=-\frac{7h}{2}$ to obtain the following discrete schemes:
\begin{align*}
&y_{n+1}=\frac{y_{n}}{2041200}+\frac{h}{2041200}\bigg[473977{f_n} +6190578f_{n+1}-14256264  f_{n+\frac{3}{2}}+21960504f_{n+2}-22333032f_{n+\frac{5}{2}}+15056670f_{ n+3}\\
&-6504408 f_{ n+\frac{7}{2}} +1635759 f_{ n+4}-182584 f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{3}{2}}=\frac{y_{n}}{89600}+\frac{h}{89600}\bigg[20759 {f_n} +287046  f_{ n+1} -581818  f_{ n+\frac{3}{2}}+936468  f_{ n+2}-958194  f_{ n+\frac{5}{2}} +647690  f_{ n+3} -280206f_{n+\frac{7}{2}}  +70533  f_{ n+4}-7878  f_{ n+\frac{9}{2}}\bigg]\\
&y_{n+2}=\frac{y_{n}}{255150}+\frac{h}{255150}\bigg[59143{f_n}+814932f_{n+1}-1601616f_{\frac{n+3}{2}}+2762856f_{n+2}-2761488f_{\frac{n+5}{2}}+1860780f_{n+3}-803952f_{\frac{n+7}{2}}+202221f_{n+4}-22576f_{\frac{n+9}{2}}\bigg]\\
&y_{n+\frac{5}{2}}=\frac{y_{n}}{2612736}+\frac{h}{2612736}\bigg[605495{f_n}+8353350  f_{ n+1}-16467450  f_{ n+\frac{3}{2}}+28962900  f_{ n+2}-27460530-f_{ n+\frac{5}{2}}+18890250  f_{ n+3}8182350  f_{ n+\frac{7}{2}}\\
&+2060325f_{ n+4}-230150f_{ n+\frac{9}{2}}\bigg]\\
&y_{n+3}=\frac{y_{n}}{2800}+\frac{h}{2800}\bigg[649{f_n}+8946f_{n+1}-17608f_{n+\frac{3}{2}}+30888f_{n+2}-28584f_{n+\frac{5}{2}}+20990f_{n+3}-8856f_{n+\frac{7}{2}}+2223f_{n+4}-248f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{7}{2}}=\frac{y_{n}}{9331200}+\frac{h}{9331200}\bigg[2162377{f_n}+29837178f_{n+1}-58823814f_{n+\frac{3}{2}}+103389804f_{n+2}-96271182f_{n+\frac{5}{2}}+73295670f_{n+3}-27390258f_{n+\frac{7}{2}}\\&+7276059f_{n+4}-816634f_{n+\frac{9}{2}}\bigg]\\
&y_{n+4}=\frac{y_{n}}{127575}+\frac{h}{127575}\bigg[29578{f_n}+407232f_{n+1}-800256f_{n+\frac{3}{2}}+1402056f_{n+2}-1294848f_{n+\frac{5}{2}}+972480f_{n+3}-317952f_{n+\frac{7}{2}}+123786f_{n+4}-11776f_{n+\frac{9}{2}}\bigg]\\
&y_{n+\frac{9}{2}}=\frac{y_{n}}{89600}+\frac{h}{89600}\bigg[20727{f_n}+288198f_{n+1}-574074f_{n+\frac{3}{2}}+1017684f_{n+2}-965682f_{n+\frac{5}{2}}+748170f_{n+3}-278478f_{n+\frac{7}{2}}+141669f_{n+4}4986f_{n+\frac{9}{2}}\bigg]
\end{align*}
\end{document}
• Welcome to tex.sx. Off topic: mathtools loads amsmath and amssymb loads amsfonts, so the preamble can be cleaned up a bit. Commented Apr 26, 2023 at 18:20
• thanks. i tremendously appreciate. Commented Apr 26, 2023 at 18:35
• It would be nice if you could edit the title of this question to something more mreaningful. Commented Apr 27, 2023 at 7:28
• @Megamind - Did you just try to "edit" my answer by, basically, deleting it entirely and replacing it with the single sentence "Thanks,I really appreciate your help."?
– Mico
Commented May 25, 2023 at 5:27
• @Megamind - If you have a new query, please post a new question. Do not ovewrite an existing query (and certainly not the associated answer!).
– Mico
Commented May 25, 2023 at 15:49

In the following code, the main change I've implemented is to user fewer \frac expressions and use more inline-fraction terms, as well as 1.5, 2.5, 3.5, and 4.5 instead of \frac{3}{2}, \frac{5}{2}, \frac{7}{2}, and \frac{9}{2}. I've also eliminated scores of useless pairs of curly braces.

There's simply no normal way to force the B matrix into the width of the textblock; I've therefore had to resort to the sledgehammer method, aka \resizebox. The result is awful, but it's the only adjustment method available since you don't want to render the matrix in landscape mode.

The following screenshot shows the first of roughly 2.5 pages that are produced by the code given below.

\documentclass[a4paper,12pt]{book}
\usepackage{graphicx}
\usepackage[inner=1.3cm, outer=1.3cm, vmargin=2cm,
bindingoffset=1cm]{geometry}
%\usepackage[utf8]{inputenc} % that's the default nowadays
\usepackage[T1]{fontenc} % <-- new
\usepackage{amssymb,mathtools}

\allowdisplaybreaks % <-- new

\begin{document}

Consider the Linear Multi-step Method:
$$\begin{split} y(x) = \alpha_{0}(x)y_n + h\big[ &\beta_{0}(x)f_n+\beta_{1}(x)f_{n+1}+\beta_{1.5}(x)f_{n+1.5}+\beta_{2}(x)f_{n+2}\\ & +\beta_{2.5}(x)f_{n+2.5}+\beta_{3}(x)f_{n+3}+\beta_{3.5}(x)f_{n+3.5} \\ &+\beta_{4}(x)f_{n+4}+\beta_{4.5}(x)f_{n+4.5}\big] \end{split}$$

$$\renewcommand\arraystretch{1.25} A=\begin{bmatrix*}[l] 1 & x_n & x_n^2 & x_n^3 & x_n^4 & x_n^5 & x_n^6 & x_n^7 & x_n^8 & x_n^9 \\ 0 & 1 & 2 x_n & 3 x_n^2 & 4 x_n^3 & 5 x_n^4 & 6 x_n^5 & 7 x_n^6 & 8 x_n^7 & 9 x_n^8 \\ 0 & 1 & 2 x_{n+1} & 3 x_{n+1}^2 & 4 x_{n+1}^3 & 5 x_{n+1}^4 & 6x_{n+1}^5 & 7 x_{n+1}^6 & 8 x_{n+1}^7 & 9 x_{n+1}^8\\ 0 & 1 & 2 x_{n+1.5} & 3 x_{n+1.5}^2 & 4 x_{n+1.5}^3 & 5 x_{n+1.5}^4 & 6 x_{n+1.5}^5 & 7 x_{n+1.5}^6 & 8 x_{n+1.5}^7 & 9 x_{n+1.5}^8 \\ 0 & 1 & 2 x_{n+2} & 3 x_{n+2}^2 & 4 x_{n+2}^3 & 5 x_{n+2}^4 & 6 x_{ n+2}^5 & 7 x_{n+2}^6 & 8 x_{n+2}^7 & 9 x_{n+2}^8\\ 0 & 1 & 2 x_{n+2.5} & 3 x_{n+2.5}^2 & 4 x_{n+2.5}^3 & 5 x_{n+2.5}^4 & 6 x_{n+2.5}^5 & 7 x_{n+2.5}^6 & 8 x_{n+2.5}^7 & 9 x_{n+2.5}^8 \\ 0 & 1 & 2 x_{n+3} & 3 x_{n+3}^2 & 4 x_{n+3}^3 & 5 x_{n+3}^4 & 6 x_{n+3}^5 & 7 x_{n+3}^6 & 8 x_{n+3}^7 & 9 x_{n+3}^8\\ 0 & 1 & 2 x_{n+3.5} & 3 x_{n+3.5}^2 & 4 x_{n+3.5}^3 & 5 x_{n+3.5}^4 & 6 x_{n+3.5}^5 & 7 x_{n+3.5}^6 & 8 x_{n+3.5}^7 & 9 x_{n+3.5}^8\\ 0 & 1 & 2 x_{n+4} & 3 x_{n+4}^2 & 4 x_{n+4}^3 & 5 x_{n+4}^4 & 6 x_{n+4}^5 & 7 x_{n+4}^6 & 8 x_{n+4}^7 & 9 x_{n+4}^8 \\ 0 & 1 & 2 x_{n+4.5} & 3 x_{n+4.5}^2 & 4 x_{n+4.5}^3 & 5 x_{n+4.5}^4 & 6 x_{n+4.5}^5 & 7 x_{n+4.5}^6 & 8 x_{n+4.5}^7 & 9 x_{n+4.5}^8 \end{bmatrix*}$$

Replacing $x_n$, $x_{n+1.5}$, $x_{n+2}$, $x_{n+2.5}$, $x_{n+3}$, $x_{n+3.5}$, $x_{n+4}$, and $x_{n+4.5}$
with $x_{n+1}-h$, $x_{n+1}+0.5h$, $x_{n+1}+h$, $x_{n+1}+1.5h$, $x_{n+1}+2h$, $x_{n+1}+2.5h$, $x_{n+1}+3h$, and $x_{n+1}+3.5h$,
we obtain:
$$\label{mat} \renewcommand\arraystretch{1.25} \medmuskip=0mu \setlength\arraycolsep{2.5pt} % default: 5pt \resizebox{0.94\textwidth}{!}{% B=\begin{bmatrix*}[l] 1 & x_{n+1}-h & (x_{n+1}-h)^2 & (x_{n+1}-h)^3 & (x_{n+1}-h)^4 & (x_{n+1}-h)^5 & (x_{n+1}-h)^6 & (x_{n+1}-h)^7 & (x_{n+1}-h)^8 & (x_{n+1}-h)^9 \\ 0 & 1 & 2 (x_{n+1}-h) & 3 (x_{n+1}-h)^2 & 4 (x_{n+1}-h)^3 & 5 (x_{n+1}-h)^4 & 6 (x_{n+1}-h)^5 & 7 (x_{n+1}-h)^6 & 8 (x_{n+1}-h)^7 & 9 (x_{n+1}-h)^8 \\ 0 & 1 & 2 x_{n+1} & 3 x_{n+1}^2 & 4 x_{n+1}^3 & 5 x_{n+1}^4 & 6x_{n+1}^5 & 7 x_{n+1}^6 & 8 x_{ n+1}^7 & 9 x_{n+1}^8\\ 0 & 1 & 2 (x_{n+1}+\frac{1}{2}h) & 3 (x_{n+1}+\frac{1}{2}h)^2 & 4 (x_{n+1}+\frac{1}{2}h)^3 & 5 (x_{n+1}+\frac{1}{2}h)^4 & 6 (x_{n+1}+\frac{1}{2}h)^5 & 7 (x_{n+1}+\frac{1}{2}h)^6 & 8 (x_{n+1}+\frac{1}{2}h)^7 & 9 (x_{n+1}+\frac{1}{2}h)^8\\ 0 & 1 & 2 (x_{n+1}+h) & 3 (x_{n+1}+h)^2 & 4 (x_{n+1}+h)^3 & 5 (x_{n+1}+h)^4 & 6 (x_{n+1}+h)^5 & 7 (x_{n+1}+h)^6 & 8 (x_{n+1}+h)^7 & 9 (x_{n+1}+h)^8 \\ 0 & 1 & 2 (x_{n+1}+1.5h) & 3 (x_{n+1}+1.5h)^2 & 4 (x_{n+1}+1.5h)^3 & 5(x_{n+1}+1.5h)^4 & 6 (x_{n+1}+1.5h)^5 & 7 (x_{n+1}+1.5h)^6 & 8 (x_{n+1}+1.5h)^7 & 9 (x_{n+1}+1.5h)^8 \\ 0 & 1 & 2 (x_{n+1}+2h) & 3 (x_{n+1}+2h)^2 & 4 (x_{n+1}+2h)^3 & 5 (x_{n+1}+2h)^4 & 6 (x_{n+1}+2h)^5 & 7 (x_{n+1}+2h)^6 & 8 (x_{n+1}+2h)^7 & 9 (x_{n+1}+2h)^8 \\ 0 & 1 & 2 (x_{n+1}+2.5h) & 3 (x_{n+1}+2.5h)^2 & 4 (x_{n+1}+2.5h)^3 & 5 (x_{n+1}+2.5h)^4 & 6 (x_{n+1}+2.5h)^5 & 7 (x_{n+1}+2.5h)^6 & 8 (x_{n+1}+2.5h)^7 & 9 (x_{n+1}+2.5h)^8 \\ 0 & 1 & 2 (x_{n+1}+3h) & 3 (x_{n+1}+3h)^2 & 4 (x_{n+1}+3h)^3 & 5 (x_{n+1}+3h)^4 & 6(x_{n+1}+3h)^5 & 7 (x_{n+1}+3h)^6 & 8 (x_{n+1}+3h)^7 & 9 (x_{n+1}+3h)^8\\ 0 & 1 & 2 (x_{n+1}+3.5h) & 3 (x_{n+1}+3.5h)^2 & 4 (x_{n+1}+3.5h)^3 & 5 (x_{n+1}+3.5h)^4 & 6 (x_{n+1}+3.5h)^5 & 7 (x_{n+1}+3.5h)^6 & 8 (x_{n+1}+3.5h)^7 & 9 (x_{n+1}+3.5h)^8 \end{bmatrix*}}$$

The new matrix is renamed $B$ for ease of reference.

The determinant of $B$ is:
$\det (B) =\tfrac{1}{64} 15380234690625 h^{36}$

The inverse of \eqref{mat} is the $C$ matrix calculated using \textbf{Maple} codes as shown in the \textbf{Appendix}.

Our only interest in the $C$ matrix is its first row. The elements are:
\begin{align*}
c_{11}&=1\\[\jot]
c_{12}&=-(160 x_{n+1}^9+2520 h x_{n+1}^8+16560 h^2 x_{n+1}^7+58800 h^3 x_{n+1}^6+121842 h^4 x_{n+1}^5+147735 h^5 x_{n+1}^4\\
&\quad+98010 h^6 {x_{n+1sing}^3}+28350 h^7 x_{n+1}^2-473977 h^9)/(2041200 h^8)\\[\jot]
c_{13}&=(320 x_{n+1}^9+4680 h x_{n+1}^8+27360 h^2 x_{n+1}^7+78960 h^3 x_{n+1}^6+102564 h^4 x_{n+1}^5-9135 h^5 x_{n+1}^4\\
&\quad-197940 h^6 x_{n+1}^3-237330 h^7 x_{n+1}^2-113400 h^8 x_{n+1}+343921 h^9)/(113400 h^8)\\[\jot]
c_{14}&=-(1120 x_{n+1}^9+15750 h x_{n+1}^8+86760 h^2 x_{n+1}^7+225750 h^3 x_{n+1}^6+223524 h^4 x_{n+1}^5\\
&\quad-171675 h^5 x_{n+1}^4-578340 h^6 x_{n+1}^3-396900 h^7 x_{n+1}^2+594011 h^9)/(85050 h^8)\\[\jot]
c_{15}&=(560 x_{n+1}^9+7560 h x_{n+1}^8+39240 h^2 x_{n+1}^7+92400 h^3 x_{n+1}^6+68607 h^4 x_{n+1}^5\\
&\quad-101745 h^5 {x_{ n+1}^4}-210735 h^6 x_{n+1}^3-99225 h^7 x_{n+1}^2+203338 h^9)/(18900 h^8) \\[\jot]
c_{16}&=-(1120 x_{n+1}^9+14490 h x_{n+1}^8+70920 h^2 x_{n+1}^7+152250 h^3 x_{n+1}^6+84924 h^4 x_{n+1}^5\\
&\quad-191205 h^5 x_{n+1}^4-310380 h^6 x_{n+1}^3-132300 h^7 x_{n+1}^2+310181 h^9)/(28350 h^8)\\[\jot]
c_{17}&=(2240 x_{n+1}^9+27720 h x_{n+1}^8+128160 h^2 x_{n+1}^7+253680 h^3 x_{n+1}^6+109116 h^4 x_{n+1}^5\\
&\quad-336735 h^5 x_{n+1}^4-487620 h^6 x_{n+1}^3-198450 h^7 x_{n+1}^2+501889 h^9)/(68040 h^8)\\[\jot]
c_{18}&=-(160 x_{n+1}^9+1890 h x_{n+1}^8+8280 h^2 x_{n+1}^7+15330 h^3 x_{n+1}^6+5292 h^4 x_{n+1}^5\\
&\quad-21105 h^5 x_{n+1}^4-28620 h^6 x_{n+1}^3-11340 h^7 x_{n+1}^2+30113 h^9)/(9450 h^8)\\[\jot]
c_{19}&=(1120 x_{n+1}^9+12600 h x_{n+1}^8+52560 h^2 x_{n+1}^7+92400 h^3 x_{n+1}^6+26334 h^4 x_{n+1}^5\\
&\quad-130725 h^5 x_{n+1}^4-169890 h^6 x_{n+1}^3-66150 h^7 x_{n+1}^2+181751 h^9)/(226800 h^8)\\[\jot]
c_{(10)}&=-(160 x_{n+1}^9+1710 h x_{n+1}^8+6840 h^2 x_{n+1}^7+11550 h^3 x_{n+1}^6+2772 h^4 x_{n+1}^5\\
&\quad-16695 h^5 x_{n+1}^4-21060 h^6 x_{n+1}^3-8100 h^7x_{n+1}^2+22823 h^9)/(255150h^8)
\end{align*}

From the above, we obtain the following continuous coefficients. This is obtained through
Maple Codes as shown in the \textbf{Appendix}:
\begin{align*}
\alpha_0(x)    &=y_n\\
h\beta_0(x)    &=\bigl(-160 w^9-2520 h w^8-16560 h^2 w^7-58800 h^3 w^6-121842 h^4 w^5-147735 h^5 w^4\\
&\quad-98010 h^6 w^3-28350 h^7 w^2+473977 h^9\bigr)\big/\bigl(2041200h^8\bigr)\\[\jot]
h\beta_1(x)    &=\bigl(320 w^9+4680 h w^8+27360 h^2 w^7+78960 h^3 w^6+102564 h^4 w^5-9135 h^5 w^4\\
&\quad-197940 h^6 w^3-237330 h^7 w^2-113400 h^8 w+343921 h^9\bigr)\big/\bigl(113400h^8\bigr)\\[\jot]
h\beta_{1.5}(x)&=\bigl(-1120 w^9-15750 h w^8-86760 h^2 w^7-225750 h^3 w^6-223524 h^4 w^5+171675 h^5 w^4\\
&\quad+578340 h^6 w^3+396900 h^7 w^2-594011 h^9\bigr)\big/\bigl(85050h^8\bigr)\\[\jot]
h\beta_2(x)    &=\bigl(560 w^9+7560 h w^8+39240 h^2 w^7+92400 h^3 w^6+68607 h^4 w^5-101745 h^5 w^4\\
&\quad-210735 h^6 w^3-99225 h^7 w^2+203338 h^9\bigr)\big/\bigl(18900h^8\bigr)\\[\jot]
h\beta_{2.5}(x)&=\bigl(-1120 w^9-14490 h {w^8}-70920 h^2 w^7-152250 h^3 w^6-84924 h^4 w^5+191205 h^5 w^4\\
&\quad+310380 h^6 w^3+132300 h^7 w^2-310181 h^9\bigr)\big/\bigl(28350h^8\bigr)\\[\jot]
h\beta_3(x)    &=\bigl(2240 w^9+27720 h w^8+128160 h^2 w^7+253680 h^3 w^6+109116 h^4 w^5-336735 h^5 w^4\\
&\quad-487620 h^6 w^3-198450 h^7 w^2+501889 h^9\bigr)\big/\bigl(68040h^8\bigr)\\[\jot]
h\beta_{3.5}(x)&=\bigl(-160 w^9-1890 h w^8-8280 h^2 w^7-15330 h^3 w^6-5292 h^4 w^5+21105 h^5 w^4\\
&\quad+28620 h^6 w^3+11340 h^7 w^2-30113 h^9\bigr)\big/\bigl(9450h^8\bigr)\\[\jot]
h\beta_4(x)    &=\bigl(1120 w^9+12600 h w^8+52560 h^2 w^7+92400 h^3 w^6+26334 h^4 w^5-130725 h^5 w^4\\
&\quad-169890 h^6 w^3-66150 h^7 w^2+181751 h^9\bigr)\big/\bigl(226800h^8\bigr)\\[\jot]
h\beta_{4.5}(x)&=\bigl(-160 w^9-1710 h w^8-6840 h^2 w^7-11550 h^3 w^6-2772 h^4 w^5+16695 h^5 w^4\\
&\quad+21060 h^6 w^3+8100 h^7 w^2-22823 h^9\bigr)\big/\bigl(255150h^8\bigr)
\end{align*}

Evaluating the above at $w=0$, $w=-0.5h$, $w=-h$, $w=-1.5h$, $w=-2h$, $w=-2.5h$,
$w=-3$, and $w=-3.5h$ to obtain the following discrete schemes:
\begin{align*}
y_{n+1}  &=\frac{y_n}{2041200}+\frac{h}{2041200}\Bigl[473977{f_n} +6190578f_{n+1}-14256264 f_{n+1.5}+21960504f_{n+2}\\
y_{n+1.5}&=\frac{y_n}{89600}+\frac{h}{89600}\Bigl[20759 {f_n} +287046 f_{n+1} -581818 f_{n+1.5}+936468 f_{n+2}\\
&\quad-958194 f_{n+2.5} +647690 f_{n+3} -280206f_{n+3.5}  +70533 f_{n+4}-7878 f_{n+4.5}\smash[t]{\Bigr]}\\
y_{n+2}  &=\frac{y_n}{255150}+\frac{h}{255150}\Bigl[59143{f_n}+814932f_{n+1}-1601616f_{n+1.5}+2762856f_{n+2}\\