Matrix and vector Sym objects

Mikkel Meyer Andersen and Søren Højsgaard

2023-01-12

Source: vignettes/sym-matrix-vector.Rmd

Matrices

Define a character matrix (covariance matrix from a certain \(AR(1)\)):

N <- 3
L1 <- diag(1, 1 + N)
L1[cbind(1+(1:N), 1:N)] <- "-alpha"
L1s <- as.Sym(L1)
L1s
## Yacas matrix:
##      [,1]   [,2]   [,3]   [,4]
## [1,] 1      0      0      0   
## [2,] -alpha 1      0      0   
## [3,] 0      -alpha 1      0   
## [4,] 0      0      -alpha 1

Now, this can be converted to a Sym object:

L1s <- as.Sym(L1)
L1s
## Yacas matrix:
##      [,1]   [,2]   [,3]   [,4]
## [1,] 1      0      0      0   
## [2,] -alpha 1      0      0   
## [3,] 0      -alpha 1      0   
## [4,] 0      0      -alpha 1

Operations can be performed:

L1s + 4
## Yacas matrix:
##      [,1]      [,2]      [,3]      [,4]
## [1,] 5         4         4         4   
## [2,] 4 - alpha 5         4         4   
## [3,] 4         4 - alpha 5         4   
## [4,] 4         4         4 - alpha 5
tmp <- L1s^4
tmp
## Yacas matrix:
##      [,1]                               [,2]                       [,3]      
## [1,] 1                                  0                          0         
## [2,] -4 * alpha                         1                          0         
## [3,] 3 * alpha^2 - -3 * alpha^2         -4 * alpha                 1         
## [4,] -(alpha^3 + alpha * (3 * alpha^2)) 3 * alpha^2 - -3 * alpha^2 -4 * alpha
##      [,4]
## [1,] 0   
## [2,] 0   
## [3,] 0   
## [4,] 1
## Yacas matrix:
##      [,1]         [,2]        [,3]       [,4]
## [1,] 1            0           0          0   
## [2,] -4 * alpha   1           0          0   
## [3,] 6 * alpha^2  -4 * alpha  1          0   
## [4,] -4 * alpha^3 6 * alpha^2 -4 * alpha 1

Or the concentration matrix \(K=L L'\) can be found:

K1s <- Simplify(L1s * Transpose(L1s))
K1s
## Yacas matrix:
##      [,1]   [,2]        [,3]        [,4]       
## [1,] 1      -alpha      0           0          
## [2,] -alpha alpha^2 + 1 -alpha      0          
## [3,] 0      -alpha      alpha^2 + 1 -alpha     
## [4,] 0      0           -alpha      alpha^2 + 1

This can be converted to \(\LaTeX\):

TeXForm(K1s)
## [1] "\\left( \\begin{array}{cccc} 1 &  - \\alpha  & 0 & 0 \\\\  - \\alpha  & \\alpha  ^{2} + 1 &  - \\alpha  & 0 \\\\ 0 &  - \\alpha  & \\alpha  ^{2} + 1 &  - \\alpha  \\\\ 0 & 0 &  - \\alpha  & \\alpha  ^{2} + 1 \\end{array} \\right)"

Which look like this:

cat("\\[ K_1 = ", TeXForm(K1s), " \\]", sep = "")

\[ K_1 = \left( \begin{array}{cccc} 1 & - \alpha & 0 & 0 \\ - \alpha & \alpha ^{2} + 1 & - \alpha & 0 \\ 0 & - \alpha & \alpha ^{2} + 1 & - \alpha \\ 0 & 0 & - \alpha & \alpha ^{2} + 1 \end{array} \right) \]

Vectors

Similar can be done for vectors:

x <- paste0("x", 1:2)
xs <- as.Sym(x)
xs
## Yacas vector:
## [1] x1 x2

And matrix-vector multiplication (or matrix-matrix multiplication):

A <- matrix(paste0(paste0("a", 1:2), rep(1:2, each = 2)), 2, 2)
As <- as.Sym(A)
As
## Yacas matrix:
##      [,1] [,2]
## [1,] a11  a12 
## [2,] a21  a22
As*xs
## Yacas vector:
## [1] a11 * x1 + a12 * x2 a21 * x1 + a22 * x2
As*As
## Yacas matrix:
##      [,1]                  [,2]                 
## [1,] a11^2 + a12 * a21     a11 * a12 + a12 * a22
## [2,] a21 * a11 + a22 * a21 a21 * a12 + a22^2

Eval 

xs
## Yacas vector:
## [1] x1 x2
Eval(xs, list(x1 = 2, x2 = 3))
## [1] 2 3
As
## Yacas matrix:
##      [,1] [,2]
## [1,] a11  a12 
## [2,] a21  a22
Eval(As, list(a11 = 11, a12 = 12, a21 = 21, a22 = 22))
##      [,1] [,2]
## [1,]   11   12
## [2,]   21   22

Disabling functionality

The functionality can be disabled as follows:

Ryacas_options("module_matvec_enabled")
## [1] TRUE
As
## Yacas matrix:
##      [,1] [,2]
## [1,] a11  a12 
## [2,] a21  a22
Ryacas_options(module_matvec_enabled = FALSE)
As
## yacas_expression(list(list(a11, a12), list(a21, a22)))