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S 19 Complex numbers. Matrices. Functions of two variables. Series

# Graph of a function of two variables. Example: f: (x,y)-> sin(x)+cos(2y)

# theta, phi e d  indicate the position of the eye: the direction of the look
# projected on the horizontal plane, its inclination relative to that plane and the
# distance of the eye from the center of the box.

# The following picture shows the meaning of theta and phi:

#                                  (see S11 too)

# If I want change the margin size I can use MARG(0.2) (reduce it to 0.2 inch) or MARG(…)

f = function(x,y) sin(x)+cos(2*y)
# I choose the size of the domain and the amount of grid lines
x = seq(-pi,pi, len=20); y = seq(-pi,pi, len=20)
# I create the "table" that x and y values associate z values
z = outer(x,y,f)
MARG(0.2); persp(x,y,z,phi=60,theta=30,d=20,col="yellow",ticktype="detailed",cex.axis=0.8,cex.lab=0.8)
# The range of z is chosen automatically. I can change it if I add in "persp" (e.g.)  …,zlim=c(-4,4),… [see
# If I want no color, and reduce the distance                                                 the 2nd graph]
MARG(0.2); persp(x,y,z,phi=60,theta=30,d=2,col=NULL,ticktype="detailed",cex.axis=0.8,cex.lab=0.8)
dev.new()                     # If I want open a new window (to retain the last graph)
#       If I want to choose the size of the window I can use  BF=…; HF=…; NW()  instead
# With "expand" I can apply an expansion to the z coordinates
MARG(0.2); persp(x,y,z,phi=60,theta=30,d=20,col="cyan",ticktype="detailed",expand=0.5,cex.axis=0.8,cex.lab=0.8)
# (I must introduce  scale=FALSE  if I want a monometric system)
# without "ticktype":
dev.new(); MARG(0.2); persp(x,y,z,phi=60,theta=30,d=20,col="green", expand=1.5,cex.axis=0.8,cex.lab=0.8)

 # To have the chart without the box I add: ",box=FALSE". # I can add pictures (with trans3d and pmat), if I have # given a name to the graph (here, fig): x = y = seq(-pi,pi, len=20); z = outer(x,y,f) MARG(0.2) fig = persp(x,y,z,phi=45,theta=65,d=20,col=NULL,box=FALSE) lines (trans3d(x=c(-pi,pi),y=0,z=0,pmat=fig), col="red",lwd=2) lines (trans3d(y=c(-pi,pi),x=0,z=0,pmat=fig), col="red",lwd=2) lines (trans3d(z=c(-1,1),x=0,y=0, pmat=fig), col="red",lwd=2) # I add ,lty=3 or =2 if I want dashed axes
# With "shade" I can produce a shadow effect: BF=3; HF=3; NW(); MARG(0.2) persp(x,y,z,phi=60,theta=30,d=20,col="yellow",ticktype="detailed", cex.axis=0.8,cex.lab=0.8,shade=0.2) # The contour lines (I increase "len" to have smoother lines) of the same "f" (see): x = seq(-pi,pi, len=200); y = seq(-pi,pi, len=200); z = outer(x,y,f) MARG(0.4); contour(x,y,z, levels = c(-1,-0.5,0,0.5,1) ,cex.axis=0.8,cex.lab=0.8) gridh( c(-pi/2,0,pi/2) ); gridv( c(-pi/2,0,pi/2) ) # On the right if we use nlevels=6 instead of levels=…: # But the contour lines can be obtained with the standard commands (see): PLANE(-3,3, -3,3); g = function(x,y) f(x,y)-k k=0; CUR(g,"black"); for(k in c(-1,-0.5)) CUR(g,"red"); for(k in c(1,0.5)) CUR(g,"blue") # If I want an empty box I must put in z (4 times) the base and use "zlim". See here: x = c(-pi,pi); y = c(-pi,pi) # extremes of x and y z0 = c(-4,4); u = rep(z0[1],4); z=array(u,dim=c(2,2)) # z0: range of 3rd dimension BF=4; HF=4 NW(); MARG(0.2) persp(x,y,z,phi=60,theta=30,d=20,col="yellow",ticktype="detailed",cex.axis=0.8,cex.lab=0.8,zlim=z0) NW(); MARG(0.2) persp(x,y,z,phi=60,theta=30,d=20,ticktype="detailed",cex.axis=0.8,cex.lab=0.8,zlim=z0) # # MINF2(F, a,b, c,d) finds where F (of 2 variables) has a minimum in [a,b]×[c,d], and its value; # MAXF2(F, a,b, c,d) finds the maximum. The computer does nothing but speed up a process that # I could do directly with zoom (see what we said on "maxmin"). # Let f be the same function as above. f = function(x,y) sin(x)+cos(2*y) BF=3.5; HF=3; NW() x = seq(-pi,pi, len=20); y = seq(-pi,pi, len=20); z = outer(x,y,f) MARG(0.2); G=persp(x,y,z,phi=60,theta=30,d=20,col="cyan",ticktype="detailed",expand=0.5,cex.axis=0.8,cex.lab=0.8) # maximun in [0,3]×[-1,2]: Q = MAXF2(f, 0,3, -1,2) # ~ max in x,y F(max) [1] 1.570796e+00 4.476588e-09 2.000000e+00 # Max in the point (π/2,0) where the value of f is 2 points(trans3d(Q[1],Q[2],Q[3], pmat = G), col="red", pch=16) P = MINF2(f, -3,0, -3,0) # ~ min in x,y F(min) [1] -1.570796 -1.570796 -2.000000 # Min in the point (-π/2,-π/2) where the value of f is -2 points(trans3d(P[1],P[2],P[3], pmat = G), col="blue", pch=16) # # Another example (figure above right): h = function(x,y) x*y*exp(-x^4-y^4) x = y = seq(-2,2, len=31); z = outer(x,y,h) BF=3.5; HF=3; NW() MARG(0.2); G=persp(x,y,z,phi=40,theta=30,d=20,col="cyan",ticktype="detailed",expand=0.5,cex.axis=0.8,cex.lab=0.8) V = MAXF2(h, -1.5,-0.5, -1,0.5) # ~ max in x,y F(max) [1] -0.7071068 -0.7071068 0.3032653 V[1]^2 # 0.5 Maximum in (-1/√2,-1/√2) where the value of f is 0.3032653 points(trans3d(V[1],V[2],V[3], pmat = G), col="red", pch=16) # # When a curve is not defined in a rectangle [a,b]×[c,d] we must integrate different # forms of representation. The study of (x,y) → x^2-log(x+y): # See here for partial derivatives and hessian # To go deep: F = function(x,y) { r = sqrt(x^2+y^2); ifelse(r==0, 10, 10 * sin(r)/r) } x1 = -10; x2 = 10; y1 = -10; y2 = 10; z1=-3; z2=10 x=seq(x1,x2,len=31); y=seq(y1,y2,len=31); z = outer(x,y,F) BF=4; HF=3 NW(); MARG(0.3) G=persp(x,y,z,phi=30,theta=30,d=2,col="lightblue",ticktype="detailed",cex.axis=0.8, cex.lab=0.8,expand=0.5,zlim=c(z1,z2)) xE = c(-10,10); xy = expand.grid(xE, xE) points(trans3d(xy[,1], xy[,2], 6, pmat = G), col="red", pch =16) lines (trans3d(x, y=10, z= 6 + sin(x), pmat = G), col="green") f = function(x) { r = sqrt(x^2); ifelse(r==0, 10, 10*sin(r)/r) } r1 = maxmin(f,5,10); r1 # 7.725252 t=seq(0, 2*pi, len=201); xr=r1*cos(t); yr=r1*sin(t) lines (trans3d(xr, yr, F(xr,yr), pmat = G), col="brown",lwd=2) # the corresponding curve levels (they are 2) f(r1) # 1.283746 x = seq(-10,10, len=400); y = seq(-10,10, len=400); z = outer(x,y,F) BF=3; HF=3; NW(); MARG(0.4); contour(x,y,z, levels = 1.283, cex.axis=0.8,cex.lab=0.8) # or: f = function(x,y) F(x,y)-k; PLANE(-10,10, -10,10); k=1.2837; CURVE(f,"brown") # A curve in space (5 laps of a propeller) t = seq(0,10*pi,len=1000) z0 = c(0,10*pi); u = rep(z0[1],4) # I put the box base in u z = array(u,dim=c(2,2)); x = c(-2,2); y = c(-2,2) F = persp(x,y,z,theta=30,phi=20,scale=TRUE,zlim=z0,xlim=x,ylim=y,d=1) lines(trans3d(cos(t),sin(t),t,pmat=F),col="red") # To have it without the box I would have added in persp: border="white",box=FALSE # See here how to obtatin the figure below in the center: # See here for the figure on the right (a conformal map): for other references to # complex numbers see K3 # See here for sequences, series and Fourier series. # See also Math & Music (slides and references) # For figure above and the our space (which is non-Euclidean) see here and # Walking on the balls (see "animazione" and "approfondimenti") # The matrices (see) are table of data that can be introduced as arrays: A = array(data=c(2,6,9), dim=c(1,3)) nrow(A); ncol(A); A; t(A) # [1] 1 # [1] 3 # [,1] [,2] [,3] A # [1,] 2 6 9 # [,1] t(A): the transpose of A # [1,] 2 # [2,] 6 # [3,] 9 B = array(data=c(1,5,8),dim=c(3,1) ); B # [,1] # [1,] 1 # [2,] 5 # [3,] 8 C = array(data=c(1,2, 5,3, 7,8), dim=c(2,3) ); C # [,1] [,2] [,3] the data are introduced by columns # [1,] 1 5 7 # [2,] 2 3 8 E = array(data=c(1,2, 5,3), dim=c(2,2) ); E # I don't use D: it is used for derivatives # [,1] [,2] # [1,] 1 5 # [2,] 2 3 # After the name I can put the indices to identify an element: B[3]; E[1,2] # 8 5 A %*% B # %*% multiplies two matrices (1st row * 1st col, # [,1] # 2nd row * 1st col, …) # [1,] 104 # C %*% E is not defined B %*% A # I can compute t(A)+B and t(A)-B because t(A) and # [,1] [,2] [,3] # B have the same number of row and col # [1,] 2 6 9 # [2,] 10 30 45 # [3,] 16 48 72 H = array(data=c(2,1,-1, 1,1,3, 0,9,2), dim=c(3,3) ); H; H %*% H %*% H # H^3 # [,1] [,2] [,3] # [1,] 2 1 0 # [2,] 1 1 0 # [3,] -1 3 2 # [,1] [,2] [,3] # [1,] 13 8 0 # [2,] 8 5 0 # [3,] 2 19 8 M = INV(E) # Inverse: the matrix M such that M %*% E (and E %*% M) is the fraction( M ) # identity matrix (diagonal entries equal to 1, other to 0) # [,1] [,2] # [1,] -3/7 5/7 # [2,] 2/7 -1/7 fraction(E %*% M) # [,1] [,2] # [1,] 1 0 # [2,] 0 1 det(E) # the determinant of the matrix E (see) # [1] -7 # An alternative method to solve linear systems. An example: # x+3y=3 AND 4x+2y=7, that we can already solve: S = c(1,3,3, 4,2,7); eqSystem(S) # [1] 1.5 0.5 # I put the coefficients col by col ma = array(data=c(1,4, 3,2),dim=c(2,2)); no = array(data(3,7),dim=c(2,1)); ma; no # [,1] [,2] # [1,] 1 3 # [2,] 4 2 # [,1] # [1,] "3" # [2,] "7" solve(ma,no) # [,1] # [1,] 1.5 # [2,] 0.5 # An alternative method to calculate scalar (or dot) product. An example: v1 = c(1,-2,2); v2 = c(-4,1,2); prods(v1,v2) # [1] -2 v1 %*% v2; drop(v1 %*% v2) # drop the dimensions of the array # [,1] # [1,] -2 # [1] -2 # For the use of arrays in statistics see here. Other examples of use