Author: Matthew Fidler
Date: 2026-05-13

# Lotri Motivation

#### Matthew Fidler

#### 2026-05-13

# Motivation

This was made to allow people (like me) to specify lower triangular matrices similar to the domain specific language implemented in nlmixr. Originally I had it included in `RxODE`, but thought it may have more general applicability, so I separated it into a new package.

For me, specifying the matrices in this way is easier than specifying them using R’s default matrix. For instance to fully specify a simple 2x2 matrix, in R you specify:

    mat <- matrix(c(1, 0.5, 0.5, 1),nrow=2,ncol=2,dimnames=list(c("a", "b"), c("a", "b")))

With `lotri`, you simply specify:

    library(lotri)
    library(microbenchmark)
    library(ggplot2)

    mat <- lotri(a+b ~ c(1,
                         0.5, 1))

I find it more legible and easier to specify, especially if you have a more complex matrix. For instance with the more complex matrix:

    mat <- lotri({
        a+b ~ c(1,
                0.5, 1)
        c ~ 1
        d +e ~ c(1,
                 0.5, 1)
    })

To fully specify this in base R you would need to use:

    mat <- matrix(c(1, 0.5, 0, 0, 0,
                    0.5, 1, 0, 0, 0,
                    0, 0, 1, 0, 0,
                    0, 0, 0, 1, 0.5,
                    0, 0, 0, 0.5, 1),
                  nrow=5, ncol=5,
                  dimnames= list(c("a", "b", "c", "d", "e"), c("a", "b", "c", "d", "e")))

Of course with the excellent `Matrix` package this is a bit easier:

    library(Matrix)
    mat <- matrix(c(1, 0.5, 0.5, 1),nrow=2,ncol=2,dimnames=list(c("a", "b"), c("a", "b")))
    mat <- bdiag(list(mat, matrix(1), mat))
    ## Convert back to standard matrix
    mat <- as.matrix(mat)
    ##
    dimnames(mat) <- list(c("a", "b", "c", "d", "e"), c("a", "b", "c", "d", "e"))

Regardless, I think `lotri` is a bit easier to use.

# Creating lists of matrices with attached properties

`lotri` also allows lists of matrices to be created by conditioning on an `id` with the `|` syntax.

For example:

    mat <- lotri({
        a+b ~ c(1,
                0.5, 1) | id
        c ~ 1 | occ
        d +e ~ c(1,
                 0.5, 1) | id(lower=3, upper=2, omegaIsChol=FALSE)
    })

    print(mat)
    #> $id
    #>     d   e
    #> d 1.0 0.5
    #> e 0.5 1.0
    #>
    #> $occ
    #>   c
    #> c 1
    #>
    #> Properties: lower, upper, omegaIsChol

    print(mat$lower)
    #> $id
    #> d e
    #> 3 3
    #>
    #> $occ
    #>    c
    #> -Inf
    print(mat$upper)
    #> $id
    #> d e
    #> 2 2
    #>
    #> $occ
    #>   c
    #> Inf
    print(mat$omegaIsChol)
    #> $id
    #> [1] FALSE

This gives a list of matrix(es) conditioned on the variable after the `|`. It also can add properties to each list that can be accessible after the list of matrices is returned, as shown in the above example. To do this, you simply have to enclose the properties after the conditional variable. That is `et1 ~ id(lower=3)`.

## Combining symmetric named matrices

Now there is even a faster way to do a similar banded matrix concatenation with `lotriMat`

    testList <- list(lotri({et2 + et3 + et4 ~ c(40,
                                0.1, 20,
                                0.1, 0.1, 30)}),
                         lotri(et5 ~ 6),
                         lotri(et1+et6 ~c(0.1, 0.01, 1)),
                         matrix(c(1L, 0L, 0L, 1L), 2, 2,
                                dimnames=list(c("et7", "et8"),
                                              c("et7", "et8"))))

    matf <- function(.mats){
      .omega <- as.matrix(Matrix::bdiag(.mats))
      .d <- unlist(lapply(seq_along(.mats),
                          function(x) {
                            dimnames(.mats[[x]])[2]
                          }))
      dimnames(.omega) <- list(.d, .d)
      return(.omega)
    }

    print(matf(testList))
    #>      et2  et3  et4 et5  et1  et6 et7 et8
    #> et2 40.0  0.1  0.1   0 0.00 0.00   0   0
    #> et3  0.1 20.0  0.1   0 0.00 0.00   0   0
    #> et4  0.1  0.1 30.0   0 0.00 0.00   0   0
    #> et5  0.0  0.0  0.0   6 0.00 0.00   0   0
    #> et1  0.0  0.0  0.0   0 0.10 0.01   0   0
    #> et6  0.0  0.0  0.0   0 0.01 1.00   0   0
    #> et7  0.0  0.0  0.0   0 0.00 0.00   1   0
    #> et8  0.0  0.0  0.0   0 0.00 0.00   0   1

    print(lotriMat(testList))
    #>      et2  et3  et4 et5  et1  et6 et7 et8
    #> et2 40.0  0.1  0.1   0 0.00 0.00   0   0
    #> et3  0.1 20.0  0.1   0 0.00 0.00   0   0
    #> et4  0.1  0.1 30.0   0 0.00 0.00   0   0
    #> et5  0.0  0.0  0.0   6 0.00 0.00   0   0
    #> et1  0.0  0.0  0.0   0 0.10 0.01   0   0
    #> et6  0.0  0.0  0.0   0 0.01 1.00   0   0
    #> et7  0.0  0.0  0.0   0 0.00 0.00   1   0
    #> et8  0.0  0.0  0.0   0 0.00 0.00   0   1

    mb <- microbenchmark(matf(testList),lotriMat(testList))

    print(mb)
    #> Unit: microseconds
    #>                expr     min       lq       mean   median       uq      max
    #>      matf(testList) 954.953 1375.338 1754.76248 1813.498 1977.617 5564.065
    #>  lotriMat(testList)   2.188    4.764    9.87591    7.625   13.658   27.843
    #>  neval
    #>    100
    #>    100
    autoplot(mb)
    #> Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
    #> ℹ Please use tidy evaluation idioms with `aes()`.
    #> ℹ See also `vignette("ggplot2-in-packages")` for more information.
    #> ℹ The deprecated feature was likely used in the microbenchmark package.
    #>   Please report the issue at
    #>   <https://github.com/joshuaulrich/microbenchmark/issues/>.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.