SparseM (version 1.84-2)

matrix.csr.chol-class: Class "matrix.csr.chol" (Block Sparse Cholesky Decomposition)

Description

A class of objects returned from Ng and Peyton's (1993) block sparse Cholesky algorithm.

Objects from the Class

Objects may be created by calls of the form new("matrix.csr.chol", ...), but typically result from chol(<matrix.csr>).

Slots

nrow:

an integer, the number of rows of the original matrix, or in the linear system of equations.

nnzlindx:

Object of class numeric, number of non-zero elements in lindx

nsuper:

an integer, the number of supernodes of the decomposition

lindx:

Object of class integer, vector of integer containing, in column major order, the row subscripts of the non-zero entries in the Cholesky factor in a compressed storage format

xlindx:

Object of class integer, vector of integer of pointers for lindx

nnzl:

of class "numeric", an integer, the number of non-zero entries, including the diagonal entries, of the Cholesky factor stored in lnz

lnz:

a numeric vector of the entries of the Cholesky factor

xlnz:

an integer vector, the column pointers for the Cholesky factor stored in lnz

invp:

inverse permutation vector, integer

perm:

permutation vector, integer

xsuper:

Object of class integer, array containing the supernode partioning

det:

numeric, the determinant of the Cholesky factor

log.det:

numeric, the log determinant of the Cholesky factor

ierr:

an integer, the error flag (from Fortran's src/chol.f)

time:

numeric, unused (always 0.) currently.

Methods

as.matrix.csr

signature(x = "matrix.csr.chol", upper.tri=TRUE): to get the sparse ("matrix.csr") upper triangular matrix corresponding to the Cholesky decomposition.

backsolve

signature(r = "matrix.csr.chol"): for computing \(R^{-1} b\) when the Cholesky decomposition is \(A = R'R\).

Details

Note that the perm and notably invp maybe important to back permute rows and columns of the decompositions, see the Examples, and our chol help page.

See Also

Base R's chol and SparseM's chol, notably for examples; backsolve

Examples

x5g <- new("matrix.csr",
          ra = c(300, 130, 5, 130, 330,
                 125, 10, 5, 125, 200, 70,
                 10, 70, 121.5, 1e30),
          ja = c(1:3, 1:4, 1:4, 2:5),
          ia = c(1L, 4L, 8L, 12L, 15L, 16L),
          dimension = c(5L, 5L))
(m5g <- as.matrix(x5g)) # yes, is symmetric, and positive definite:
eigen(m5g, only.values=TRUE)$values  # all positive (but close to singular)
ch5g <- chol(x5g)
str(ch5g) # --> the slots of the "matrix.csr.chol" class
mch5g <- as.matrix.csr(ch5g)
print.table(as.matrix(mch5g), zero.print=".") # indeed upper triagonal w/ positive diagonal

## x5 has even more extreme entry at [5,5]:
x5 <- x5g; x5[5,5] <- 2.9e32
m5 <- as.matrix(x5)
(c5 <- chol(m5))# still fine, w/ [5,5] entry = 1.7e16 and other diag.entries in (9.56, 17.32)
ch5 <- chol(x5) # --> warning  "Replaced 3 tiny diagonal entries by 'Large'"
                # gave error for a while
(mmc5 <- as.matrix(as.matrix.csr(ch5)))
        # yes, these replacements were extreme, and the result is "strange'
## Solve the problem (here) specifying non-default  singularity-tuning par 'tiny':
ch5. <- chol(x5, tiny = 1e-33)
(mmc5. <- as.matrix(as.matrix.csr(ch5.))) # looks much better.
## Indeed: R'R  back-permuted *is* the original matrix x5, here m5:
(RtR <- crossprod(mmc5.)[ch5.@invp, ch5.@invp])
          all.equal(m5, RtR, tolerance = 2^-52)
stopifnot(all.equal(m5, RtR, tolerance = 1e-14)) # on F38 Linux, only need tol = 1.25e-16