Examples of Symbolic Operations

Description

Examples of symbolic operations using modified mathematical operators.

It may take a long time to complete all the examples.

Details

Japanese help file: SymbolicOps_JPH

Examples

# load modified mathemetical operators
modify_math_operators()


# Define matrices and vectors
set.seed(1701+1701)
Pn=matrix(round(100*runif(4)),2,2)
Qn=matrix(round(100*runif(4)),2,2)
Rn=matrix(round(100*runif(9)),3,3)
P=demomat(2,2,root="p")
Q=demomat(2,2,root="q")
R=demomat(3,3,root="r")
A=demomat(3,2,root="a")
B=demomat(3,2,root="b")
C=demomat(2,3,root="c")
v2=demomat(2,1,root="v", vec=1)
v3=demomat(3,1,root="v", vec=1)
one2=matrix(1,2,1) # numeric
one3=matrix(1,3,1) # numeric

# Define native R vectors
vec2=c("c1","c2")
vec3=c("c1","c2", "c3")


# transposition
t(A)

# trace
tr(P)

# diagonal elements
Diag(P)
diag(P)



# Binary Operators


# numeric objects
(vn <- 1+2-(1:2)*(2:3))
vn + Pn%*%Qn



# matrix with a scalar
1 + A
2 - v2
3 * A
B %*% 4

# vector + or - vector
v2 + v2
v2 - v2

# matrix + or - matrix
A + B
A - B

# matrix x vector
A %*% v2
t(v3) %*% A

# matrix x vector
A %*% one2
t(one3) %*% A


# matrix x matrix
P %*% Q
Q %*% P

# matrix x matrix
A %*% C
C %*% A


# multiplying the colums of a matrix
A %*% Diag(P)

# multiplying the rows of a matrix
Diag(R) %*% A



# permutation of the columns
A %*% pmat(2,1,2)

# permutation of the rows
pmat(3,1,2) %*% A



# orthogonal matrix
temp <- " cos(theta) -sin(theta)
          sin(theta) cos(theta)"
T <- cards(temp)
T <- matrix(unlist(T),2,2)
tTT <- t(T%*%T)
Eval(tTT,theta=0)
Eval(tTT,theta=0, check=0)


# planar rotation in 3D
temp <- "
1         0            0
0         cos(theta1) -sin(theta1)
0         sin(theta1) cos(theta1)
"
Tx <- matrix(unlist(cards(temp)),3,3)
temp <- "
cos(theta2)  0            -sin(theta2)
0            1            0
sin(theta2)  0           cos(theta2)
"
Ty <- matrix(unlist(cards(temp)),3,3)
temp <- "
cos(theta3)  -sin(theta3)   0
sin(theta3)  cos(theta3)    0
0            0              1
"
Tz <- matrix(unlist(cards(temp)),3,3)

# rotation in 3D
T <- Tx%*%Ty%*%Tz
T <- Simplify(T)
Tn <- Eval(T, values="theta1=1; theta2=2; theta3=3", check=0)
# see if T is orthonormal
t(Tn) %*% Tn |> round(.3)
Tn%*%t(Tn)  |> round(.3)



# Precedence of multiplication over addition
P + t(Q)%*%Q

# Not with \%p\%, \%m\%, \%T\%, \%t\%
P %p% t(Q)%*%Q



# transposition of the product of two matrices: t(PQ) == t(Q) t(P)
tPQ <- t( P%*%Q )
tQtP <- t(Q)%*%t(P)
Eval( tPQ-tQtP, P=Pn, Q=Qn)




# Hadamar product
A * B
v2 * v2

# Kronecker product
A %@% B
one2 %@% P
diag(3) %@% P
b_diag( P,P,P)


# R native vector product
vec3 * A
A * vec3
vec2 * A






# inverse matrix: 2 x 2
Pinv=Inv(P)
PinvP=Pinv %*% P
Eval(PinvP, P=Pn)

# inverse matrix: 3 x 3
Rinv=Inv(R)
RRinv=R %*% Rinv
Eval( RRinv, R=Rn ) # Cannot use demomat(3,3) here.

# inverse of a diagonal matrix
Inv(Diag(R))

# inverse of a lower triangular matrix: n x n
n <- 3
LT <- demomat(n,n,root="t", shape="lowert")
LTn <- demomat(n,n)
invLT <- Inv(LT)
invLT <- Expand(invLT)
LTinvLT <- LT%*%invLT
Simplify(Expand(LTinvLT))
Eval(LTinvLT,LT=LTn)


# inverse of the transposed matrix: Inv(t(P)) = t(Inv(P))
invtP <- Inv(t(P))
tinvP <- t(Inv(P))
Print(invtP,"\n",tinvP)
Eval(invtP-tinvP, P=Pn, Q=Qn)

# inverse of the product of two matrices: Inv(PQ) == Inv(Q) Inv(P)
invPQ <- Inv(P%*%Q)
invQinvP <- Inv(Q)%*%Inv(P)
dif <- invPQ-invQinvP
Eval(dif, P=Pn, Q=Qn)

# block diagonal matrix and its inverse
A2 <- demomat(2,2,root="a")
B2 <- demomat(2,2,root="b")
( bigAB2 <- b_diag(A2,B2) )
( invbigAB2 <- Inv(bigAB2) )
( invBigAB2 <- b_diag(Inv(A2),Inv(B2)) )
invBigAB2-invbigAB2
# Eval(invbigAB2-invBigAB2, A2=demomat(2,2),B=demomat(2,2))


# determinant
( temp <- Det(P) )
temp |>  Expand(simplify=0) |> Simplify()

# log determinant
Det(P,log=1)

# checing..
( dd <- Det(R) )
Rn <- matrix(runif(3*3),3)
Print( Eval(dd,R=Rn),det(Rn) )

( ddd <- Expand(dd) |> Expand() |> Simplify() )
Print( Eval(ddd,R=Rn),det(Rn) )



# sum of all the element
t(one3) %*% A %*% one2
Sum(A)

# column sum
t(one3) %*% A
apply( A, 2, Sum)

# row sum
A %*% one2
apply( A, 1, Sum)



# centering operator
J <- diag(3) - ("(1/3)"%*%one3%*%t(one3))

# centering of a matrix
( temp <- J %*% A )
Expand(temp)
apply( Eval( J%*%A, A=matrix(runif(3*2),3) ), 2, sum )



# sum of squared elements of a vector
t(v2) %*% v2
Sum( v2 * v2 )


# sum of squared elements of a matrix
tr(t(A) %*% A)
tr(A %*% t(A))
Sum(A * A)



# vectorization of a matrix
vec(R)


# vec and Kronecker product
# vec( X %*% Y %*% Z)  ==  ( t(Z) %@% X ) %*% vec(Y)
X=demomat(2,2,root="x"); Y=demomat(2,2,root="y"); Z=demomat(2,2,root="z")
Xn=demomat(2,2); Yn=demomat(2,2); Zn=demomat(2,2)
vecXYZ <- vec(X%*%Y%*%Z)
vecXYZ <- Expand( vecXYZ )
tZKXvY <- (t(Z)%@%X) %*% vec(Y)
Print(vecXYZ,"\n",tZKXvY)
dif <- vecXYZ - tZKXvY
Eval(dif, X=Xn, Y=Yn, Z=Zn)


# trace, vec and Kronecker product: tr(XYZ) == t(vec(t(X)) I%@%Y vec(Z)
trXYZ <- tr( X%*%Y%*%Z )
trXYZ <- Expand(trXYZ)
vXetc <- t(vec(t(X))) %*% (diag(2)%@%Y) %*% vec(Z)
vXetc <- Expand(vXetc)
(dif <- trXYZ - vXetc)



# vectorization of the lower half of a matrix
( temp <- vech(R) )
vechinv(temp)
( temp <- vech(R,1) )
vechinv(temp,1)




# Partitioned Matrix


# Fully Symbolic Partitioned Matrix
# 2 x 2 partitioned matrices
Ap=demomat(2,2,root="A" ,fullsymb=1)
Bp=demomat(2,2,root="B" ,fullsymb=1)

Print(Ap,Bp)


# Semi Symbolic Partitioned Matrix
A11=demomat(2,2,root="a11_"); A12=demomat(2,3,root="a12_")
A21=demomat(3,2,root="a21_"); A22=demomat(3,3,root="a22_")
B11=demomat(2,2,root="b11_"); B12=demomat(2,3,root="b12_")
B21=demomat(3,2,root="b21_"); B22=demomat(3,3,root="b22_")
bigA=rbind( cbind(A11,A12), cbind(A21,A22) )
# attributes(bigA)$SMdim=list(nrows=c(2,3), ncols=c(2,3))
 bigB=rbind( cbind(B11,B12), cbind(B21,B22) )
# attributes(bigB)$SMdim=list(nrows=c(2,3), ncols=c(2,3))
Printb(bigA, sep="  ")


# semi-symbolic checking
bigAp=matReplace( Ap, A11=A11, A12=A12, A21=A21, A22=A22)
Print(bigA,bigAp)


# transpose
( tAp=t(Ap) )

# semi-symbolic checking
bigtAp=matReplace( tAp, A11=A11, A12=A12, A21=A21, A22=A22)
Print(t(bigA),bigtAp)


# addition and subtraction
Ap+Bp
Ap-Bp

# semi-symbolic checking
vlist="A11=A11; A12=A12; A21=A21; A22=A22; B11=B11; B12=B12; B21=B21; B22=B22"
matReplace(Ap+Bp, values=vlist)
bigA+bigB

# multiplication
Ap %*% Bp

# semi-symbolic checking
matReplace(Ap%*%Bp, values=vlist) - bigA%*%bigB


# diagonaization
Diag(Ap)
Diag( map("Diag", diag(Ap)) )


# block triangular matrix
( bigL <- demomat(3,3,root="A", shape="lowert", fullsymb=1) )
# inverse of block triangular matrix
( invbigL <- matSweep(bigL) )

# full-symbolic checking
Expand(Simplify( bigL%*%invbigL, all=1 ), all=1 )
Expand(Simplify( invbigL%*%bigL, all=1 ), all=1 )



# invserse by matSweep (Inv)
( invAp1=Inv(Ap) )
( invAp2=Inv(Ap, 2:1) )

# full-symbolic checking
( iAA <- Simplify(Expand(invAp1 %*% Ap, all=1),all=1) )
( AiA <- Simplify(Expand(Ap %*% invAp1, all=1),all=1) )

# pick up the 12 element of inv(Ap1)%+%Ap
iAA12 <- iAA[1,2]
analyze_expr( iAA12 )

# Let A22.1=A22-A21%*%inv(A11)%*%A12 and simplify visually.
iAA12s <- gsub("A22-A21%*%inv(A11)%*%A12","A22.1", iAA12, fixed=1 )
analyze_expr( iAA12s )

# pick up the 22 element of inv(Ap1)%+%Ap
iAA22 <- iAA[2,2]
analyze_expr( iAA22 )

# Let A22.1=A22-A21%*%inv(A11)%*%A12 and simplify visually.
iAA22s <- gsub("A22-A21%*%inv(A11)%*%A12","A22.1", iAA22, fixed=1 )
analyze_expr( iAA22s )