library(Matrix)

### Matrix Products including  cross products

source(system.file("test-tools.R", package = "Matrix"))

### dimnames -- notably for matrix products
## from ../R/Auxiliaries.R :
.M.DN <- function(x) if(!is.null(dn <- dimnames(x))) dn else list(NULL,NULL)
dnIdentical <- function(x,y) identical(.M.DN(x), .M.DN(y))
chkDnProd <- function(m, M = Matrix(m), browse=FALSE) {
    ## TODO:
    ## if(browse) stopifnot <- f.unction(...)  such that it enters browser()..
    stopifnot(is.matrix(m), is(M, "Matrix"))
    ## m is  n x d  (say)

    p1 <- (tm <- t(m)) %*% m ## d x d
    p1. <- crossprod(m)
    stopifnot(dnIdentical(p1, p1.))

    t1 <- m %*% tm ## n x n
    t1. <- tcrossprod(m)
    stopifnot(dnIdentical(t1, t1.))

    ## Now the	"Matrix" ones -- should match the "matrix" above
    M0 <- M
    cat("sparse: ")
    for(sparse in c(TRUE, FALSE)) {
	cat(sparse, "; ")
	M <- as(M0, if(sparse)"sparseMatrix" else "denseMatrix")
	P1 <- (tM <- t(M)) %*% M
	P1. <- crossprod(M)
	stopifnot(dnIdentical(P1, P1.), dnIdentical(P1, p1),
		  dnIdentical(P1., crossprod(M,M)),
		  dnIdentical(P1., crossprod(M,m)),
		  dnIdentical(P1., crossprod(m,M)))

	## P1. is "symmetricMatrix" -- semantically "must have" symm.dimnames
	PP1 <- P1. %*% P1. ## still  d x d
	R <- triu(PP1);r <- as(R,"matrix") # upper - triangular
	L <- tril(PP1);l <- as(L,"matrix") # lower - triangular
	stopifnot(isSymmetric(P1.), isSymmetric(PP1),
		  is(L,"triangularMatrix"), is(R,"triangularMatrix"),
		  dnIdentical(PP1, (pp1 <- p1 %*% p1)),
		  dnIdentical(PP1, R),
		  dnIdentical(L, R))

	T1 <- M %*% tM
	T1. <- tcrossprod(M)
	stopifnot(dnIdentical(T1, T1.), dnIdentical(T1, t1),
		  dnIdentical(T1., tcrossprod(M,M)),
		  dnIdentical(T1., tcrossprod(M,m)),
		  dnIdentical(T1., tcrossprod(m,M)),
		  dnIdentical(tcrossprod(T1., tM),
			      tcrossprod(t1., tm)),
		  dnIdentical(crossprod(T1., M),
			      crossprod(t1., m)))

	## Now, *mixing*  Matrix x matrix:
	stopifnot(dnIdentical(tM %*% m, tm %*% M))

	## Symmetric and Triangular
	stopifnot(dnIdentical(PP1 %*% tM, pp1 %*% tm),
		  dnIdentical(R %*% tM, r %*% tm),
		  dnIdentical(L %*% tM, L %*% tm))
    }
    cat("\n")


    invisible(TRUE)
}

## All these are ok  {now, (2012-06-11) also for dense
(m <- matrix(c(0, 0, 2:0), 3, 5))
m00 <- m # *no* dimnames
dimnames(m) <- list(LETTERS[1:3], letters[1:5])
(m.. <- m) # has *both* dimnames
m0. <- m.0 <- m..
dimnames(m0.)[1] <- list(NULL); m0.
dimnames(m.0)[2] <- list(NULL); m.0
##
chkDnProd(m..)
chkDnProd(m0.)
chkDnProd(m.0)
chkDnProd(m00)



m5 <- 1 + as(diag(-1:4)[-5,], "dgeMatrix")
## named dimnames:
dimnames(m5) <- list(Rows= LETTERS[1:5], paste("C", 1:6, sep=""))
tr5 <- tril(m5[,-6])
m. <- as(m5, "matrix")
m5.2 <- local({t5 <- as.matrix(tr5); t5 %*% t5})
stopifnot(isValid(tr5, "dtrMatrix"),
	  dim(m5) == 5:6,
	  class(cm5 <- crossprod(m5)) == "dpoMatrix")
assert.EQ.mat(t(m5) %*% m5, as(cm5, "matrix"))
assert.EQ.mat(tr5.2 <- tr5 %*% tr5, m5.2)
stopifnot(isValid(tr5.2, "dtrMatrix"),
	  as.vector(rep(1,6) %*% cm5) == colSums(cm5),
	  as.vector(cm5 %*% rep(1,6)) == rowSums(cm5))
## uni-diagonal dtrMatrix with "wrong" entries in diagonal
## {the diagonal can be anything: because of diag = "U" it should never be used}:
tru <- Diagonal(3, x=3); tru[i.lt <- lower.tri(tru, diag=FALSE)] <- c(2,-3,4)
tru@diag <- "U" ; stopifnot(diag(trm <- as.matrix(tru)) == 1)
## TODO: Also add *upper-triangular*  *packed* case !!
stopifnot((tru %*% tru)[i.lt] ==
	  (trm %*% trm)[i.lt])

## crossprod() with numeric vector RHS and LHS
assert.EQ.mat( crossprod(rep(1,5), m5),	 rbind( colSums(m5)))
assert.EQ.mat( crossprod(rep(1,5), m.),	 rbind( colSums(m5)))
assert.EQ.mat( crossprod(m5, rep(1,5)),	 cbind( colSums(m5)))
assert.EQ.mat( crossprod(m., rep(1,5)),	 cbind( colSums(m5)))
## tcrossprod() with numeric vector RHS and LHS :
m51 <- m5[, 1, drop=FALSE] # [6 x 1]
m.1 <- m.[, 1, drop=FALSE] ; assert.EQ.mat(m51, m.1)
## The only (M . v) case
assert.EQ.mat(tcrossprod(m51, 5:1), tcrossprod(m.1, 5:1))
## The two  (v . M) cases:
assert.EQ.mat(tcrossprod(rep(1,6), m.), rbind( rowSums(m5)))# |v| = ncol(m)
assert.EQ.mat(tcrossprod(rep(1,3), m51),
              tcrossprod(rep(1,3), m.1))# ncol(m) = 1

## classes differ
tc.m5 <- m5 %*% t(m5)	 # "dge*", no dimnames (FIXME)
(tcm5 <- tcrossprod(m5)) # "dpo*"  w/ dimnames
assert.EQ.mat(tc.m5, mm5 <- as(tcm5, "matrix"))
## tcrossprod(x,y) :
assert.EQ.mat(tcrossprod(m5, m5), mm5)
assert.EQ.mat(tcrossprod(m5, m.), mm5)
assert.EQ.mat(tcrossprod(m., m5), mm5)

M50 <- m5[,FALSE, drop=FALSE]
M05 <- t(M50)
s05 <- as(M05, "sparseMatrix")
s50 <- t(s05)
assert.EQ.mat(M05, matrix(1, 0,5))
assert.EQ.mat(M50, matrix(1, 5,0))
assert.EQ.mat(tcrossprod(M50), tcrossprod(as(M50, "matrix")))
assert.EQ.mat(tcrossprod(s50), tcrossprod(as(s50, "matrix")))
assert.EQ.mat( crossprod(s50),	crossprod(as(s50, "matrix")))
stopifnot(identical( crossprod(s50), tcrossprod(s05)),
	  identical( crossprod(s05), tcrossprod(s50)))
(M00 <- crossprod(M50))## used to fail -> .Call(dgeMatrix_crossprod, x, FALSE)
stopifnot(identical(M00, tcrossprod(M05)),
	  all(M00 == t(M50) %*% M50), dim(M00) == 0)

## simple cases with 'scalars' treated as 1x1 matrices:
d <- Matrix(1:5)
d %*% 2
10 %*% t(d)
assertError(3 %*% d)		 # must give an error , similar to
assertError(5 %*% as.matrix(d))	 # -> error

## right and left "numeric" and "matrix" multiplication:
(p1 <- m5 %*% c(10, 2:6))
(p2 <- c(10, 2:5) %*% m5)
(pd1 <- m5 %*% diag(1:6))
(pd. <- m5 %*% Diagonal(x = 1:6))
(pd2 <- diag (10:6)	   %*% m5)
(pd..<- Diagonal(x = 10:6) %*% m5)
stopifnot(dim(crossprod(t(m5))) == c(5,5),
	  c(class(p1),class(p2),class(pd1),class(pd2),
	    class(pd.),class(pd..)) == "dgeMatrix")
assert.EQ.mat(p1, cbind(c(20,30,33,38,54)))
assert.EQ.mat(pd1, m. %*% diag(1:6))
assert.EQ.mat(pd2, diag(10:6) %*% m.)
assert.EQ.mat(pd., as(pd1,"matrix"))
assert.EQ.mat(pd..,as(pd2,"matrix"))

## check that 'solve' and '%*%' are inverses
set.seed(1)
A <- Matrix(rnorm(25), nc = 5)
y <- rnorm(5)
all.equal((A %*% solve(A, y))@x, y)
Atr <- new("dtrMatrix", Dim = A@Dim, x = A@x, uplo = "U")
all.equal((Atr %*% solve(Atr, y))@x, y)

### ------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
### Sparse Matrix products
### ------
## solve() for dtC*
mc <- round(chol(crossprod(A)), 2)
B <- A[1:3,] # non-square on purpose
stopifnot(all.equal(sum(rowSums(B %*% mc)), 5.82424475145))
assert.EQ.mat(tcrossprod(B, mc), as.matrix(t(tcrossprod(mc, B))))

m <- kronecker(Diagonal(2), mc)
stopifnot(is(mc, "Cholesky"),
	  is(m, "sparseMatrix"))
im <- solve(m)
round(im, 3)
itm <- solve(t(m))
iim <- solve(im) # should be ~= 'm' of course
iitm <- solve(itm)
I <- Diagonal(nrow(m))
(del <- c(mean(abs(as.numeric(im %*% m - I))),
	  mean(abs(as.numeric(m %*% im - I))),
	  mean(abs(as.numeric(im  - t(itm)))),
	  mean(abs(as.numeric( m  - iim))),
	  mean(abs(as.numeric(t(m)- iitm)))))
stopifnot(is(m, "triangularMatrix"), is(m, "sparseMatrix"),
	  is(im, "dtCMatrix"), is(itm, "dtCMatrix"), is(iitm, "dtCMatrix"),
	  del < 1e-15)

## crossprod(.,.) & tcrossprod(),  mixing dense & sparse
v <- c(0,0,2:0)
mv <- as.matrix(v) ## == cbind(v)
(V <- Matrix(v, 5,1, sparse=TRUE))
sv <- as(v, "sparseVector")
a <- as.matrix(A)
cav <-	crossprod(a,v)
tva <- tcrossprod(v,a)
assert.EQ.mat(crossprod(A, V), cav) # gave infinite recursion
assert.EQ.mat(crossprod(A,sv), cav)
assert.EQ.mat(tcrossprod( sv, A), tva)
assert.EQ.mat(tcrossprod(t(V),A), tva)

## [t]crossprod()  for  <sparsevector> . <sparsevector>  incl. one arg.:
stopifnot(isValid(s.s <- crossprod(sv,sv), "Matrix"),
	  identical(s.s, crossprod(sv)),
	  isValid(ss. <- tcrossprod(sv,sv), "sparseMatrix"),
	  identical(ss., tcrossprod(sv)))
assert.EQ.mat(s.s,  crossprod(v,v))
assert.EQ.mat(ss., tcrossprod(v,v))

dm <- Matrix(v, sparse=FALSE)
sm <- Matrix(v, sparse=TRUE)
stopifnot(
    identical4(tcrossprod(v, v), tcrossprod(mv, v),
	       tcrossprod(v,mv), tcrossprod(mv,mv))## (base R)
    ,
    validObject(d.vvt <- as(as(vvt <- tcrossprod(v, v), "denseMatrix"), "dgeMatrix"))
    ,
    identical4(d.vvt,		 tcrossprod(dm, v),
	       tcrossprod(v,dm), tcrossprod(dm,dm)) ## (v, dm) failed
    ,
    validObject(s.vvt <- as(as(vvt, "sparseMatrix"), "dgCMatrix"))
    ,
    identical(s.vvt, tcrossprod(sm,sm))
    ,
    identical3(d.vvt, tcrossprod(sm, v),
               tcrossprod(v,sm)) ## both (sm,v) and (v,sm) failed
    )
assert.EQ.mat(d.vvt, vvt)
assert.EQ.mat(s.vvt, vvt)

M <- Matrix(0:5, 2,3) ; sM <- as(M, "sparseMatrix"); m <- as(M, "matrix")
v <- 1:3; v2 <- 2:1
sv  <- as( v, "sparseVector")
sv2 <- as(v2, "sparseVector")
tvm <- tcrossprod(v, m)
assert.EQ.mat(tcrossprod( v, M), tvm)
assert.EQ.mat(tcrossprod( v,sM), tvm)
assert.EQ.mat(tcrossprod(sv,sM), tvm)
assert.EQ.mat(tcrossprod(sv, M), tvm)
assert.EQ.mat(crossprod(M, sv2), crossprod(m, v2))
stopifnot(identical(tcrossprod(v, M), v %*% t(M)),
	  identical(tcrossprod(v,sM), v %*% t(sM)),
	  identical(tcrossprod(v, M), crossprod(v, t(M))),
	  identical(tcrossprod(sv,sM), sv %*% t(sM)),
	  identical(crossprod(sM, sv2), t(sM) %*% sv2),
	  identical(crossprod(M, v2), t(M) %*% v2))

## *unit* triangular :
t1 <- new("dtTMatrix", x= c(3,7), i= 0:1, j=3:2, Dim= as.integer(c(4,4)))
## from	 0-diagonal to unit-diagonal {low-level step}:
tu <- t1 ; tu@diag <- "U"
cu <- as(tu, "dtCMatrix")
cl <- t(cu) # unit lower-triangular
cl10 <- cl %*% Diagonal(4, x=10)
assert.EQ.mat(cl10, as(cl, "matrix") %*% diag(4, x=10))
stopifnot(is(cl,"dtCMatrix"), cl@diag == "U")
(cu2 <- cu %*% cu)
cl2 <- cl %*% cl
validObject(cl2)

cu3 <- tu[-1,-1]
assert.EQ.mat(crossprod(tru, cu3),
	      crossprod(trm, as.matrix(cu3)))
## "FIXME" should return triangular ...

cl2
cu2. <- Diagonal(4) + Matrix(c(rep(0,9),14,0,0,6,0,0,0), 4,4)
D4 <- Diagonal(4, x=10:7)
stopifnot(all(cu2 == cu2.),# was wrong for ver. <= 0.999375-4
	  is(cu2, "dtCMatrix"), is(cl2, "dtCMatrix"), # triangularity preserved
	  cu2@diag == "U", cl2@diag == "U",# UNIT-triangularity preserved
	  all.equal(D4 %*% cu, D4 %*% as.matrix(cu)),
	  all.equal(cu %*% D4, as.matrix(cu) %*% D4),
	  isValid(su <- crossprod(cu), "dsCMatrix"),
	  all(D4 %*% su == D4 %*% as.mat(su)),
	  all(su %*% D4 == as.mat(su) %*% D4),
	  identical(t(cl2), cu2), # !!
	  identical( crossprod(cu), Matrix( crossprod(as.matrix(cu)),sparse=TRUE)),
	  identical(tcrossprod(cu), Matrix(tcrossprod(as.matrix(cu)),sparse=TRUE)))
tr8 <- kronecker(rbind(c(2,0),c(1,4)), cl2)
T8 <- tr8 %*% (tr8/2) # triangularity preserved?
T8.2 <- (T8 %*% T8) / 4
stopifnot(is(T8, "triangularMatrix"), T8@uplo == "L", is(T8.2, "dtCMatrix"))
mr8 <- as(tr8,"matrix")
m8. <- (mr8 %*% mr8 %*% mr8 %*% mr8)/16
assert.EQ.mat(T8.2, m8.)

data(KNex); mm <- KNex$mm
M <- mm[1:500, 1:200]
MT <- as(M, "TsparseMatrix")
cpr   <- t(mm) %*% mm
cpr.  <- crossprod(mm)
cpr.. <- crossprod(mm, mm)
stopifnot(is(cpr., "symmetricMatrix"),
	  identical3(cpr, as(cpr., class(cpr)), cpr..))
## with dimnames:
m <- Matrix(c(0, 0, 2:0), 3, 5)
dimnames(m) <- list(LETTERS[1:3], letters[1:5])
m
p1 <- t(m) %*% m
(p1. <- crossprod(m))
t1 <- m %*% t(m)
(t1. <- tcrossprod(m))
stopifnot(isSymmetric(p1.),
	  isSymmetric(t1.),
	  identical(p1, as(p1., class(p1))),
	  identical(t1, as(t1., class(t1))),
	  identical(dimnames(p1), dimnames(p1.)),
	  identical(dimnames(p1), list(colnames(m), colnames(m))),
	  identical(dimnames(t1), dimnames(t1.))
	  )

showMethods("%*%", class=class(M))

v1 <- rep(1, ncol(M))
str(r <-  M %*% Matrix(v1))
str(rT <- MT %*% Matrix(v1))
stopifnot(identical(r, rT))
str(r. <- M %*% as.matrix(v1))
stopifnot(identical4(r, r., rT, M %*% as(v1, "matrix")))

v2 <- rep(1,nrow(M))
r2 <- t(Matrix(v2)) %*% M
r2T <- v2 %*% MT
str(r2. <- v2 %*% M)
stopifnot(identical3(r2, r2., t(as(v2, "matrix")) %*% M))


###------------------------------------------------------------------
### Close to singular matrix W
### (from  micEconAids/tests/aids.R ... priceIndex = "S" )
(load(system.file("external", "symW.rda", package="Matrix"))) # "symW"
stopifnot(is(symW, "symmetricMatrix"))
n <- nrow(symW)
I <- .sparseDiagonal(n, shape="g")
S <- as(symW, "matrix")
sis <- solve(S,    S)
## solve(<dsCMatrix>, <sparseMatrix>) when Cholmod fails was buggy for *long*:
SIS <- solve(symW, symW)
iw <- solve(symW)
iss <- iw %*% symW
##                                     nb-mm3
assert.EQ.(I, drop0(sis), tol = 1e-9)# 2.6e-10
assert.EQ.(I, SIS,        tol = 1e-7)# 8.2e-9
assert.EQ.(I, iss,        tol = 4e-4)# 3.3e-5
## solve(<dsCMatrix>, <dense..>) :
I <- diag(nr=n)
SIS <- solve(symW, as(symW,"denseMatrix"))
iw  <- solve(symW, I)
iss <- iw %*% symW
assert.EQ.mat(SIS, I, tol = 1e-7, giveRE=TRUE)
assert.EQ.mat(iss, I, tol = 4e-4, giveRE=TRUE)
rm(SIS,iss)

WW <- as(symW, "generalMatrix") # the one that gave problems
IW <- solve(WW)
class(I1 <- IW %*% WW)# "dge" or "dgC" (!)
class(I2 <- WW %*% IW)
## these two were wrong for for M.._1.0-13:
assert.EQ.(as(I1,"matrix"), I, tol = 1e-4)
assert.EQ.(as(I2,"matrix"), I, tol = 7e-7)

## now slightly perturb WW (and hence break exact symmetry
set.seed(131); ii <- sample(length(WW), size= 100)
WW[ii] <- WW[ii] * (1 + 1e-7*runif(100))
SW. <- symmpart(WW)
SW2 <- Matrix:::forceSymmetric(WW)
stopifnot(all.equal(as(SW.,"matrix"),
                    as(SW2,"matrix"), tol = 1e-7))
(ch <- all.equal(WW, as(SW.,"dgCMatrix"), tolerance =0))
stopifnot(is.character(ch), length(ch) == 1)## had length(.)  2  previously
IW <- solve(WW)
class(I1 <- IW %*% WW)# "dge" or "dgC" (!)
class(I2 <- WW %*% IW)
I <- diag(nr=nrow(WW))
stopifnot(all.equal(as(I1,"matrix"), I, check.attributes=FALSE, tolerance = 1e-4),
          ## "Mean relative difference: 3.296549e-05"  (or "1.999949" for Matrix_1.0-13 !!!)
          all.equal(as(I2,"matrix"), I, check.attributes=FALSE)) #default tol gives "1" for M.._1.0-13

if(doExtras) {
    print(kappa(WW)) ## [1] 5.129463e+12
    print(rcond(WW)) ## [1] 6.216103e-14
    ## Warning message: rcond(.) via sparse -> dense coercion
}

class(Iw. <- solve(SW.))# FIXME? should be "symmetric" but is not
class(Iw2 <- solve(SW2))# FIXME? should be "symmetric" but is not
class(IW. <- as(Iw., "denseMatrix"))
class(IW2 <- as(Iw2, "denseMatrix"))

### The next two were wrong for very long, too
assert.EQ.(I, as.matrix(IW. %*% SW.), tol= 4e-4)
assert.EQ.(I, as.matrix(IW2 %*% SW2), tol= 4e-4)
dIW <- as(IW, "denseMatrix")
assert.EQ.(dIW, IW., tol= 4e-4)
assert.EQ.(dIW, IW2, tol= 8e-4)

##------------------------------------------------------------------



## Sparse Cov.matrices from  Harri Kiiveri @ CSIRO
a <- matrix(0,5,5)
a[1,2] <- a[2,3] <- a[3,4] <- a[4,5] <- 1
a <- a + t(a) + 2*diag(5)
b <- as(a, "dsCMatrix") ## ok, but we recommend to use Matrix() ``almost always'' :
(b. <- Matrix(a, sparse = TRUE))
stopifnot(identical(b, b.))

## calculate conditional variance matrix ( vars 3 4 5 given 1 2 )
(B2 <- b[1:2, 1:2])
bb <- b[1:2, 3:5]
stopifnot(is(B2, "dsCMatrix"), # symmetric indexing keeps symmetry
	  identical(as.mat(bb), rbind(0, c(1,0,0))),
	  ## TODO: use fully-sparse cholmod_spsolve() based solution :
	  is(z.s <- solve(B2, bb), "sparseMatrix"))
assert.EQ.mat(B2 %*% z.s, as(bb, "matrix"))
## -> dense RHS and dense result
z. <- solve(as(B2, "dgCMatrix"), bb)# now *sparse*
z  <- solve( B2, as(bb,"dgeMatrix"))
stopifnot(is(z., "sparseMatrix"),
          all.equal(z, as(z.,"denseMatrix")))
## finish calculating conditional variance matrix
v <- b[3:5,3:5] - crossprod(bb,z)
stopifnot(all.equal(as.mat(v),
		    matrix(c(4/3, 1:0, 1,2,1, 0:2), 3), tol = 1e-14))


###--- "logical" Matrices : ---------------------

##__ FIXME __ now works for lsparse* and nsparse* but not yet for  lge* and nge* !

## Robert's Example, a bit more readable
fromTo <- rbind(c(2,10),
		c(3, 9))
N <- 10
nrFT <- nrow(fromTo)
rowi <- rep.int(1:nrFT, fromTo[,2]-fromTo[,1] + 1) - 1:1
coli <- unlist(lapply(1:nrFT, function(x) fromTo[x,1]:fromTo[x,2])) - 1:1

## "n" --- nonzero pattern Matrices
sM  <- new("ngTMatrix", i = rowi, j=coli, Dim=as.integer(c(N,N)))
sM # nice

sm <- as(sM, "matrix")
sM %*% sM
assert.EQ.mat(sM %*% sM,	sm %*% sm)
assert.EQ.mat(t(sM) %*% sM,
	      (t(sm) %*% sm) > 0, tol=0)
crossprod(sM)
tcrossprod(sM)
stopifnot(identical(as( crossprod(sM), "ngCMatrix"), t(sM) %*%	 sM),
	  identical(as(tcrossprod(sM), "ngCMatrix"),  sM  %*% t(sM)))

assert.EQ.mat( crossprod(sM),  crossprod(sm) > 0)
assert.EQ.mat(tcrossprod(sM), as(tcrossprod(sm),"matrix") > 0)

## "l" --- logical Matrices -- use usual 0/1 arithmetic
nsM <- sM
sM  <- as(sM, "lMatrix")
sm <- as(sM, "matrix")
stopifnot(identical(sm, as.matrix(nsM)))
sM %*% sM
assert.EQ.mat(sM %*% sM,	sm %*% sm)
assert.EQ.mat(t(sM) %*% sM,
	      t(sm) %*% sm, tol=0)
crossprod(sM)
tcrossprod(sM)
stopifnot(identical( crossprod(sM), as(t(sM) %*% sM, "symmetricMatrix")),
	  identical(tcrossprod(sM), forceSymmetric(sM %*% t(sM))))
assert.EQ.mat( crossprod(sM),  crossprod(sm))
assert.EQ.mat(tcrossprod(sM), as(tcrossprod(sm),"matrix"))
dm <- as(sm, "denseMatrix")
## the following 6 products (dm o sM) all failed up to 2013-09-03
isValid(dm %*% sM,            "CsparseMatrix")## failed {missing coercion}
isValid(crossprod (dm ,   sM),"CsparseMatrix")
isValid(tcrossprod(dm ,   sM),"CsparseMatrix")
dm[2,1] <- TRUE # no longer triangular
isValid(           dm %*% sM, "CsparseMatrix")
isValid(crossprod (dm ,   sM),"CsparseMatrix")
isValid(tcrossprod(dm ,   sM),"CsparseMatrix")

## A sparse example - with *integer* matrix:
M <- Matrix(cbind(c(1,0,-2,0,0,0,0,0,2.2,0),
		  c(2,0,0,1,0), 0, 0, c(0,0,8,0,0),0))
t(M)
(-4:5) %*% M
stopifnot(as.vector(print(t(M %*% 1:6))) ==
	  c(as(M,"matrix") %*% 1:6))
(M.M <- crossprod(M))
MM. <- tcrossprod(M)
stopifnot(class(MM.) == "dsCMatrix",
	  class(M.M) == "dsCMatrix")

M3 <- Matrix(c(rep(c(2,0),4),3), 3,3, sparse=TRUE)
I3 <- as(Diagonal(3), "CsparseMatrix")
m3 <- as.matrix(M3)
iM3 <- solve(m3)
stopifnot(all.equal(unname(iM3), matrix(c(3/2,0,-1,0,1/2,0,-1,0,1), 3)))
assert.EQ.mat(solve(as(M3, "sparseMatrix")), iM3)
assert.EQ.mat(solve(I3,I3), diag(3))
assert.EQ.mat(solve(M3, I3), iM3)# was wrong because I3 is unit-diagonal
assert.EQ.mat(solve(m3, I3), iM3)# gave infinite recursion in (<=) 0.999375-10

isValid(tru %*% I3,		"triangularMatrix")
isValid(crossprod(tru, I3),	"triangularMatrix")
isValid(crossprod(I3, tru),	"triangularMatrix")
isValid(tcrossprod(I3, tru),	"triangularMatrix")

## even simpler
m <- matrix(0, 4,7); m[c(1, 3, 6, 9, 11, 22, 27)] <- 1
(mm <- Matrix(m))
(cm <- Matrix(crossprod(m)))
stopifnot(identical(crossprod(mm), cm))
(tm1 <- Matrix(tcrossprod(m))) #-> had bug in 'Matrix()' !
(tm2 <- tcrossprod(mm))
Im2 <- solve(tm2[-4,-4])
P <- as(as.integer(c(4,1,3,2)),"pMatrix")
p <- as(P, "matrix")
P %*% mm
assertError(mm %*% P) # dimension mismatch
assertError(m  %*% P) # ditto
assertError(crossprod(t(mm), P)) # ditto
stopifnot(isValid(tm1, "dsCMatrix"),
	  all.equal(tm1, tm2, tolerance =1e-15),
	  identical(drop0(Im2 %*% tm2[1:3,]), Matrix(cbind(diag(3),0))),
	  identical(p, as.matrix(P)),
	  identical(P %*% m, as.matrix(P) %*% m),
	  all(P %*% mm	==  P %*% m),
	  all(P %*% mm	-   P %*% m == 0),
	  all(t(mm) %*% P ==  t(m) %*% P),
	  identical(crossprod(m, P),
		    crossprod(mm, P)),
	  TRUE)

d <- function(m) as(m,"dsparseMatrix")
IM1 <- as(c(3,1,2), "indMatrix")
IM2 <- as(c(1,2,1), "indMatrix")
assert.EQ.Mat(crossprod(  IM1,   IM2),
              crossprod(d(IM1),d(IM2)), tol=0)# failed at first

set.seed(123)
for(n in 1:250) {
    n1 <- 2 + rpois(1, 10)
    n2 <- 2 + rpois(1, 10)
    N <- rpois(1, 25)
    ii <- seq_len(N + min(n1,n2))
    IM1 <- as(c(sample(n1), sample(n1, N, replace=TRUE))[ii], "indMatrix")
    IM2 <- as(c(sample(n2), sample(n2, N, replace=TRUE))[ii], "indMatrix")
    ## stopifnot(identical(crossprod(  IM1,    IM2),
    ##                     crossprod(d(IM1), d(IM2))))
    if(!identical(C1 <- crossprod(  IM1,    IM2 ),
                  CC <- crossprod(d(IM1), d(IM2))) &&
       !all(C1 == CC)) {
        cat("The two crossprod()s differ: C1 - CC =\n")
        print(C1 - CC)
        stop("The two crossprod()s differ!")
    } else if(n %% 25 == 0) cat(n, " ")
}; cat("\n")


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