determinare stabilità metodo SD

[prezzemolina86]

lo schema numerico è
\( \displaystyle \)v^'_l=(1/delta^2)*(-1/12*v_(l-2)+4/3*v_(l-1)-5/2*v_(l)+4/3*v_(l+1)-1/12*v_(l-2)) \( \displaystyle \)

utilizzando l analisi con serie di fourier
\( \displaystyle \)a(theta)=(-1/12)*exp(-2i theta)+4/3*exp(-i theta)-5/2+4/3*exp(i theta)+(-1/12)*exp(-2i theta) \( \displaystyle \lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt}{t}{r}{a}{\mathsf{{\quad\text{or}\quad}}}{m}{\quad\text{and}\quad}{o}\in{s}{e}{n}{o}{e}{\cos{{e}}}{n}{i}{e}{u}{t}{i}{l}{i}{z}{z}{\quad\text{and}\quad}{o}{l}{a}{f{{\quad\text{or}\quad}}}\mu{l}{a}{d}{i}{d}{u}{p}{l}{i}{c}{a}{z}{i}{o}\ne\partial{\cos{{e}}}{n}{o}{r}{a}{g{{g{{i}}}}}{u}{n}{g{{o}}}{i}{l}{s}{e}{g{{u}}}{e}{n}{t}{e}{r}{i}{s}{\underline{{t}}}{a}\to\lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt} \) \( \displaystyle {a}{\left(\theta\right)}={\left(-\frac{{1}}{{3}}\right)}\cdot{\left({{\cos}}^{{2}}\theta+{8}{\cos{\theta}}-{7}\right)} \) \( \displaystyle \lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt}{e}{q}{u}\in{d}{i}????{c}{h}{e}{c}{o}{n}{c}{l}{u}{s}{i}{o}\ne{p}{o}{s}{s}{o}{d}{a}{r}{e}?\lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt}{s}{u}{d}{e}{g{{l}}}{i}{a}{p}{p}{u}{n}{t}{i}\in{\vec{{e}}}{i}{l}{r}{i}{s}{\underline{{t}}}{a}\to{r}{a}{g{{g{{i}}}}}{u}{n}\toè{i}{l}{s}{e}{g{{u}}}{e}{n}{t}{e}\lt{b}\frac{{r}}{\gt} \) \( \displaystyle {a}{\left(\theta\right)}={\left(-\frac{{1}}{{2}}\right)}\cdot{\left({1}-{\cos{\theta}}\right)}{\left({7}{\cos{\theta}}\right)} \)

tra l'altro c'è scritto che \( \displaystyle {7}{\cos{\theta}} \) è sempre positivo e non capisco il perchè??