# Problem 13:
#
pr = polyroot( c( 1.25, 2, 1 ) )
print(pr)


# Problem 14:
#
pr = polyroot( c( -4, 9, 9 ) )
print(pr)


# Problem 15:
#
pr = polyroot( c( 1.25, 1, 1 ) )
print(pr)


# Problem 16:
#
pr = polyroot( c( 6.25, 4, 1 ) )
print(pr)


# Problem 18:
#
pr = polyroot( c( 5, 4, 1 ) )
print(pr)


# Problem 19:
#
pr = polyroot( c( 5, -2, 1 ) )
print(pr)


# Problem 20:
#
A = matrix( c( 1/2, sqrt(3)/2, -sqrt(3)/2, 1/2 ), nrow=2, ncol=2, byrow=T )
b=c( 2, -4)
x = solve( A, b )
print(x)


# Problem 21:
#
pr = polyroot( c( 1.25, 1, 1 ) )
print(pr)


# Problem 22:
#
pr = polyroot( c( 2, 2, 1 ) )
print(pr)

A = matrix( c( 1/2, sqrt(3)/2, -sqrt(3)/2, 1/2 ), nrow=2, ncol=2, byrow=T )
b=c(2, 4)
x = solve( A, b )
print(x)


# Problem 23:
#
pr = polyroot( c( 2, -1, 3 ) )
print(pr)

u = function(t){
    2 * exp(t/6) * cos( ( sqrt(23)/6 ) * t ) - (2 / sqrt(23)) * exp(t/6) * sin( ( sqrt(23)/6 ) * t )
}

# Looks like the first time |u| = 10 is when 10 < t < 11:
#
library(stats)
res = uniroot( function(t){ u(t) + 10 }, c(10.0, 11.0) )
t = res$root
print(sprintf('u(t)=-10 when t= %f', t))

ts = seq(0, 15, length.out=500 )
us = u(ts)

#postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_3_prob_23_plot.eps", onefile=FALSE, horizontal=FALSE)
plot(ts, us, type='l', xlab='t', ylab='u(t)')
abline(h=c(-10, 10), col='blue')
abline(v=t, col='black')
grid()
#dev.off()


# Problem 24:
#
pr = polyroot( c( 7, 2, 5) )
print(pr)

u = function(t){
    2 * exp(-t/5) * cos( ( sqrt(34)/5 ) * t ) + (7/sqrt(34)) * exp(-t/5) * sin( ( sqrt(34)/5 ) *t )
}
ts = seq(5, 30, length.out=1000 )
#ts = seq(12.5, 15, length.out=1000 )
us = u(ts)

# Looks like |u(t)| < 0.1 for all times with t > T when 14.0 < T < 15.0
#
library(stats)
res = uniroot( function(t){ u(t) + 0.1 }, c(14.0, 15.0) )
t = res$root
print(sprintf('|u(t)|<0.1 for all t > T when T= %f', t))

#postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_3_prob_24_plot.eps", onefile=FALSE, horizontal=FALSE)
plot(ts, us, type='l', xlab='t', ylab='u(t)')
abline(h=c(-0.1, 0.1), col='blue')
abline(v=t, col='black')
grid()
#dev.off()


# Problem 25:
#
pr = polyroot( c( 6, 2, 1) )
print(pr)

u = function(t, a=1.0){
    2 * exp(-t) * cos( sqrt(5) * t ) + ((a+2)/sqrt(5)) * exp(-t) * sin( sqrt(5) * t )
}

alpha = -2 * ( cos(sqrt(5)) / sin(sqrt(5)) ) * sqrt(5) - 2
print(sprintf('alpha= %f', alpha))

ts = seq(0, 5, length.out=1000 )
us = u(ts, a=alpha)
#postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_3_prob_25_part_b_plot.eps", onefile=FALSE, horizontal=FALSE)
plot(ts, us, type='l', xlab='t', ylab='u(t)')
grid()
#dev.off()

ts = seq( 0, pi, length.out=1000 ) / sqrt(5)
ys = tan( sqrt(5) * ts )
plot( ts, ys, type='l' )
grid()



# Problem 26:
#
y = function(t, a=1){
    exp(-a*t) * cos(t) + a * exp(-a*t) * sin(t)
}

ts = seq(0, 12.5, length.out=1000 )
ys_a_1 = y(ts, a=1.0)
ys_a_1_over_4 = y(ts, a=1/4)
ys_a_1_over_2 = y(ts, a=1/2)
ys_a_2 = y(ts, a=2)
y_min = min( c(-0.1, ys_a_1, ys_a_1_over_4, ys_a_1_over_2, ys_a_2) )
y_max = max( c(+0.1, ys_a_1, ys_a_1_over_4, ys_a_1_over_2, ys_a_2) )
#postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_3_prob_26_part_b_plot.eps", onefile=FALSE, horizontal=FALSE)
plot(ts, ys_a_1, type='l', xlab='t', ylab='y(t)', ylim=c(y_min, y_max), col='black')
points(ts, ys_a_1_over_4, type='l', col='blue')
points(ts, ys_a_1_over_2, type='l', col='green')
points(ts, ys_a_2, type='l', col='red')
abline(h=c(-0.1, +0.1), col='black')
grid()
legend( 'topright', c('a=1', 'a=1/4', 'a=1/2', 'a=2'), lwd=2, lty=c(1, 1, 1, 1), col=c('black', 'blue', 'green', 'red') )
#dev.off()


# Find the smallest T such that ly(t)|<0.1 for all t > T
#
library(stats)
all_as = c()
all_ts = c()

a=1.0
res = uniroot( function(t){ y(t, a=a) - 0.1 }, c(1.5, 2.5) )
t = res$root
print(sprintf('a= %f: |y(t)|<0.1 for all t > T when T= %f', a, t)) 
all_as = c( all_as, a )
all_ts = c( all_ts, t )

a=1/4
res = uniroot( function(t){ y(t, a=a) - 0.1 }, c(6, 8) )
t = res$root
print(sprintf('a= %f: |y(t)|<0.1 for all t > T when T= %f', a, t))
all_as = c( all_as, a )
all_ts = c( all_ts, t )

a=1/2
res = uniroot( function(t){ y(t, a=a) + 0.1 }, c(3.5, 4.5) )
t = res$root
print(sprintf('a= %f: |y(t)|<0.1 for all t > T when T= %f', a, t))
all_as = c( all_as, a )
all_ts = c( all_ts, t )

a=2
res = uniroot( function(t){ y(t, a=a) - 0.1 }, c(1.0, 2.0) )
t = res$root
print(sprintf('a= %f: |y(t)|<0.1 for all t > T when T= %f', a, t))
all_as = c( all_as, a )
all_ts = c( all_ts, t )

indxs = sort.list(all_as)
all_as = all_as[indxs]
all_ts = all_ts[indxs]

#postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_3_prob_26_part_c_plot.eps", onefile= FALSE, horizontal= FALSE)
plot( all_as, all_ts, type='l', xlab='a', ylab='T' )
grid()
#dev.off()


# Problem 36:
#
pr = polyroot( c( 2, 3, 1 ) )
print(pr)


# Problem 37:
#
pr = polyroot( c( 1.25, 2, 1 ) )
print(pr)


# Problem 40:
#
pr = polyroot( c( 5, -2, 1 ) )
print(pr)


# Problem 42:
#
pr = polyroot( c( 10, 6, 1 ) )
print(pr)


# Problem 46:
#
pr = polyroot( c( 1, 1, 1 ) )
print(pr)