#
#  The $ tells MATALG to turn off paging, which is critical if
#  we are to easily create an input data file with which to test it.
#
$
#
#  Open a transcript file called "output"
#
d
output
#
#  Display the help screen.
#
h
#
#  Enter a 3 by 3 matrix A, 
#  estimate its reciprocal condition number,
#  change the 2,3 entry to 6,
#  estimate the reciprocal condition number again,
#  and compute B=A*transpose(A).
#
e a 3,3
 1,  2,  3
 4,  5, 10
 7,  8,  9

= rcond(a)
 
a(2,3) = 6

= rcond(a)
 
b = a * trans(a)
#
#  Now compute the singular value decomposition of A,
#  and save the factors as U, S and V.
#
m
svd
a
u
s
v
#
#  Now make sure that the singular value decomposition
#  can reproduce the original matrix A.
#
w = u * s * trans(v)
#
#  Now discard the variables U, S, V and W to save space.
#
k u s v w
#
#  Now compute the QR factorization of A
#
m qr a
q
r
#
#  And now verify that A=Q*R
#
= q * r
#
#  Now let's compute the PLU factorization of A
#
m plu a
l
u
p
#
#  And let's verify that P*A=L*U, that is, A = P*L*U
#
= p * l * u
#
#  Now let's decide to save the value of the formula we
#  just calculated in a matrix C.  And then let's print C
#  to show that it's been saved.
#
s c

t c
#
#  Now let's compute the characteristic polynomial of A,
#  and save the coefficients in CVEC.
#
p a
cvec
#
#  Let's see what symbols are now stored in the program.
#
t all
#
#  Now clear out ALL user defined variables.
#
i
yes

#
#  Enter a matrix A,
#
#  Enter a vector V (in this case, the first row of A), 
#
#  Compute B, the Householder matrix which will,
#  for each column of A, reflect the portion perpendicular to V.
#
#  Compute B*A.  
#

e a 3,3
 1,  3, -4
 4,  1,  1
 7, -1,  0

e v 3, 1
1, 4, 7

b = house(v)
c = b*a
#
#  Compute the polynomial with roots 1, 2 and 3.
#  Store its coefficients in w.
#  Evaluate the polynomial at evenly spaced points
#  using the L command.
#
i
yes

e v 3 1
1,2,3

p v
w

x = 0
= poly(w,x)

l x, 0, 4, 8

#
#  Compute the eigenvalues and eigenvectors of a matrix A.
#
i
yes

e a,4,4

 17,  -8, -12, 14
 46, -22, -35, 41
 -2,   1,   5, -4
  4,  -2,  -2,  3

l = eval(a)
p = evec(a)
b = inv(p)*a*p
#
#  Now let's show how we could save the 2 x 2 central block 
#  of B using the "U" (partition) command.
#
u b
2 3 0
2 3 0
d
no

#
#  Show how repeated evaluation works.
#
i
yes

x = 0.734
x = 4*x*(1-x)
r 10
#
#  Check how MATALG will handle the inverse of a singular
#  matrix.
#
i
yes

e a 2,2
5,10
1,2

= inv(a)
#
#  Make some other simple checks.
#
i
yes


b = id(2)
c = zeros(3)
d = ran(c)
#
#  Check the matrix norms.
#
e d 3,3
  1, 2, 3
  4, 5, 6
  7, 8, 9
= norm0(d)
= norm1(d)
= norm2(d)
#
#  Load the 5 by 5 Hilbert matrix, H,
#  have MATALG compute the inverse, G,
#  compute H*G.
#  Load the 5 by 5 Inverse Hilbert matrix, F,
#  compute H*F, notice that even now we don't get I!
#  compute F-G.
#
h = hilbert(5)
g = inv(h)
= h*g
f = hilbinv(5)
= h*f
= f-g
#
#  Check out the GCF and LCM functions.
#
= gcd(20,12)
= gcd(20,17)
= gcd(20,-12)
= gcd(20,0)
= lcm(12,4)
= lcm(12,5)
= lcm(12,12)
= lcm(12,0)
= lcm(12,-6)
#
#  Try out the HELP facility.
#
?
Commands
Enter a variable



#
#  Try lots of scalar functions.
#
x = 1.0

y = abs(-1)
y = acos(0)
y = asin(1)
y = atan(1)
y = atan2(10,10)
y = cos(pi)
y = cosd(45)
y = cosh(1)
y = e
y = eps
y = exp(1)
y = int(pi)
y = log(10)
y = log2(1000)
y = log10(100)
y = max(1,10)
y = min(1,10)
y = neg(10)
y = nint(1.9)
y = pi
y = ran(x)
y = round(1)
y = round(eps)
y = round(0.00000001)
rtol = 0.0001
y = round(eps)
y = seed
y = sin(pi)
y = sind(45)
y = sinh(1)
y = sqrt(100)
y = step(10)
y = tan(pi)
y = tand(45)
y = tanh(1)
y = 10!
#
#  Quit
#
quit
yes