### Testing positive definite matrices

library(Matrix)

h9 <- Hilbert(9)
stopifnot(c(0,0) == dim(Hilbert(0)),
          c(9,9) == dim(h9))
str(h9)
all.equal(c(determinant(h9)$modulus), -96.7369456, tol= 2e-8)
stopifnot(0 == length(h9@factors))# nothing yet
round(ch9 <- chol(h9), 3) ## round() preserves 'triangular' !
str(f9 <- as(chol(h9), "dtrMatrix"))
## h9 now has factorization
stopifnot(names(h9@factors) == "Cholesky",
          all.equal(rcond(h9), 9.0938e-13),
          all.equal(rcond(f9), 9.1272e-7, tol = 1e-6))# more precision fails
str(h9)# has 'factors'
options(digits=4)
(cf9 <- crossprod(f9))# looks the same as  h9 :
stopifnot(all.equal(as.matrix(h9),
                    as.matrix(cf9), tol= 1e-15))

h9. <- round(h9, 2)# actually loses pos.def. "slightly"
h9.p <- as(h9., "dppMatrix")
h4  <- h9.[1:4, 1:4] # this and the next
h9.[1,1] <- 10       # had failed in 0.995-14
h9.p. <- as(h9., "dppMatrix")
h9.p[1,1] <- 10 # failed in 0.995-14

stopifnot(is(h9., "symmetricMatrix"),
          is(h9.p, "symmetricMatrix"),
          is(h4,   "symmetricMatrix"))

h9.p[1,2] <- 99 #-> becomes "dgeMatrix"

str(hp9 <- as(h9, "dppMatrix"))# packed
stopifnot(is(thp9 <- t(hp9), "dppMatrix"))

hs <- as(hp9, "dspMatrix")
hs@x <- 1/hp9@x # is not pos.def. anymore
validObject(hs)
stopifnot(diag(hs) == seq(1, by = 2, length = 9))

s9 <- solve(hp9, seq(nrow(hp9)))
signif(t(s9)/10000, 4)# only rounded numbers are platform-independent
(I9 <- hp9 %*% s9)
m9 <- matrix(1:9, dimnames = list(NULL,NULL))
stopifnot(all.equal(m9, as.matrix(I9), tol = 2e-9))

cat('Time elapsed: ', proc.time(),'\n') # for ``statistical reasons''