| Title: | Sparse Linear Algebra |
|---|---|
| Description: | Some basic linear algebra functionality for sparse matrices is provided: including Cholesky decomposition and backsolving as well as standard R subsetting and Kronecker products. |
| Authors: | Roger Koenker [cre, aut], Pin Tian Ng [ctb] (Contributions to Sparse QR code), Yousef Saad [ctb] (author of sparskit2), Ben Shaby [ctb] (author of chol2csr), Martin Maechler [ctb] (chol() tweaks; S4, <https://orcid.org/0000-0002-8685-9910>) |
| Maintainer: | Roger Koenker <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.84-2 |
| Built: | 2024-11-14 04:59:11 UTC |
| Source: | https://github.com/cran/SparseM |
A virtual class needed by the "matrix.csc.hb" class
A virtual Class: No objects may be created from it.
No methods defined with class "character or NULL" in the signature.
One of the four matrices from the least-squares solution of problems in surveying that were used by Michael Saunders and Chris Paige in the testing of LSQR
data(lsq)data(lsq)
A list of class matrix.csc.hb or matrix.ssc.hb depending
on how the coefficient matrix is stored with the following components:
ra component of the csc or ssc format of the coefficient matrix, X.
ja component of the csc or ssc format of the coefficient matrix, X.
ia component of the csc or ssc format of the coefficient matrix, X.
ra component of the right-hand-side, y, if stored in csc or ssc format; right-hand-side stored in dense vector or matrix otherwise.
ja component of the right-hand-side, y, if stored in csc or ssc format; a null vector otherwise.
ia component of the right-hand-side, y, if stored in csc or ssc format; a null vector otherwise.
vector of the exact solutions, b, if they exist; a null vector o therwise.
vector of the initial guess of the solutions if they exist; a null vector otherwise.
dimenson of the coefficient matrix, X.
dimenson of the right-hand-side, y.
storage mode of the right-hand-side; can be full storage or same format as the coefficient matrix.
Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
Matrix Market, https://math.nist.gov/MatrixMarket/data/Harwell-Boeing/lsq/lsq.html
read.matrix.hb
data(lsq) class(lsq) # -> [1] "matrix.csc.hb" model.matrix(lsq)->X class(X) # -> "matrix.csr" dim(X) # -> [1] 1850 712 y <- model.response(lsq) # extract the rhs length(y) # [1] 1850data(lsq) class(lsq) # -> [1] "matrix.csc.hb" model.matrix(lsq)->X class(X) # -> "matrix.csr" dim(X) # -> [1] 1850 712 y <- model.response(lsq) # extract the rhs length(y) # [1] 1850
Class for sparse matrices stored in coordinate aka
“triplet” format, storing for each non-zero entry x[i,j]
the triplet (i,j, x[i,j]), in slots (ia, ja, ra).
Objects can be created by calls of the form new("matrix.coo", ...),
but typically rather by as.matrix.coo().
ra:Object of class numeric, a real array of nnz elements containing the non-zero
elements of A.
ja:Object of class integer, an integer array of nnz elements containing the column
indices of the elements stored in ‘ra’.
ia:Object of class integer, an integer array of nnz elements containing the row
indices of the elements stored in ‘ra’.
dimension:Object of class integer, dimension of the matrix
signature(x = "matrix.coo"): ...
signature(x = "matrix.coo"): ...
signature(x = "matrix.coo"): ...
signature(x = "matrix.coo"): ...
try( new("matrix.coo") ) # fails currently {FIXME!} # the 1x1 matrix [0] ## correponds to base matrix() mcoo <- new("matrix.coo", ra=NA_real_, ia = 1L, ja = 1L, dimension = c(1L, 1L)) mcoo # currently *does* print but wrongly: as.matrix.csr(<matrix.coo>) fails to keep NA !! co2 <- new("matrix.coo", ra = c(-Inf, -2, 3, Inf), ia = c(1L,1:3), ja = 2L*(1:4), dimension = c(7L, 12L)) co2 # works fine (as has no NA) ## Sparse Diagonal (from "numeric"): as(1:5, "matrix.diag.csr") # a sparse version of diag(1:5)try( new("matrix.coo") ) # fails currently {FIXME!} # the 1x1 matrix [0] ## correponds to base matrix() mcoo <- new("matrix.coo", ra=NA_real_, ia = 1L, ja = 1L, dimension = c(1L, 1L)) mcoo # currently *does* print but wrongly: as.matrix.csr(<matrix.coo>) fails to keep NA !! co2 <- new("matrix.coo", ra = c(-Inf, -2, 3, Inf), ia = c(1L,1:3), ja = 2L*(1:4), dimension = c(7L, 12L)) co2 # works fine (as has no NA) ## Sparse Diagonal (from "numeric"): as(1:5, "matrix.diag.csr") # a sparse version of diag(1:5)
A class for sparse matrices stored in compressed sparse column ('csc') format.
Objects can be created by calls of the form new("matrix.csc", ...).
ra:Object of class numeric, a real array of nnz elements containing the non-zero
elements of A, stored in column order. Thus, if i<j, all elements
of column i precede elements from column j. The order of elements
within the column is immaterial.
ja:Object of class integer, an integer array of nnz elements containing the row
indices of the elements stored in ‘ra’.
ia:Object of class integer, an integer array of n+1 elements containing pointers to
the beginning of each column in the arrays ‘ra’ and ‘ja’. Thus
‘ia[i]’ indicates the position in the arrays ‘ra’ and ‘ja’
where the ith column begins. The last, (n+1)st, element of ‘ia’
indicates where the n+1 column would start, if it existed.
dimension:Object of class integer, dimension of the matrix
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
signature(x = "matrix.csc"): ...
cscM <- as.matrix.csc(as(diag(4:1), "matrix.csr")) cscM str(cscM) stopifnot(identical(dim(cscM), c(4L, 4L)))cscM <- as.matrix.csc(as(diag(4:1), "matrix.csr")) cscM str(cscM) stopifnot(identical(dim(cscM), c(4L, 4L)))
A class consisting of the coefficient matrix and the right-hand-side of a linear system of equations, initial guess of the solution and the exact solutions if they exist stored in external files using the Harwell-Boeing format.
Objects can be created by calls of the form new("matrix.csc.hb", ...).
ra:Object of class numeric, ra component of the csc or ssc format of the coefficient matrix, X.
ja:Object of class integer, ja component of the csc or ssc format of the coefficient matrix, X.
ia:Object of class numeric, ia component of the csc or ssc format of the coefficient matrix, X.
rhs.ra:Object of class numeric, ra component of the right-hand-side, y, if stored in csc or
ssc format; right-hand-side stored in dense vector or matrix otherwise.
guess:Object of class numeric or NULL vector of the initial guess of the solutions if they exist;
a null vector otherwise.
xexact:Object of class numeric or NULL vector of the exact solutions, b, if they exist; a null vector otherwise.
dimension:Object of class integer, dimenson of the coefficient matrix, X.
rhs.dim:Object of class integer, dimenson of the right-hand-side, y.
rhs.mode:Object of class character or NULL storage mode of the right-hand-side; can be full storage or
same format as the coefficient matrix.
signature(object = "matrix.csc.hb"): ...
signature(object = "matrix.csc.hb"):
show() the object, notably also when auto-printing.
model.matrix, model.response,
read.matrix.hb, matrix.ssc.hb-class
A class for sparse matrices stored in compressed sparse row ('csr') format.
Objects can be created by calls of the form new("matrix.csr", ...)
and coerced from various other formats. Coercion of integer scalars
and vectors into identity matrices and diagonal matrices respectively
is accomplished by as(x,"matrix.diag.csr") which generates an
object that has all the rights and responsibilties of the "matrix.csr" class.
The default "matrix.csr" object, i.e., new("matrix.csr"), is
a scalar (1 by 1) matrix with element 0.
ra:Object of class numeric, a real array of nnz elements containing the non-zero
elements of A, stored in row order. Thus, if , all elements
of row i precede elements from row j. The order of elements
within the rows is immaterial.
ja:Object of class integer, an integer array of nnz elements containing the column
indices of the elements stored in ra.
ia:A class integer array of n+1 elements containing pointers to
the beginning of each row in the arrays ra and ja. Thus
‘ia[i]’ indicates the position in the arrays ra and ja
where the ith row begins. The last, (n+1)st, element of ia
indicates where the n+1 row would start, if it existed.
dimension:An integer, dimension of the matrix
signature(x = "matrix.csr", y = "matrix.csr"): ...
signature(x = "matrix.csr", y = "matrix"): ...
signature(x = "matrix.csr", y = "numeric"): ...
signature(x = "matrix", y = "matrix.csr"): ...
signature(x = "numeric", y = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(a = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.csr"): ...
signature(x = "matrix.diag.csr"): ...
new("matrix.csr") # the 1x1 matrix [0] new("matrix.diag.csr") # the 'same' as(1:5, "matrix.diag.csr") # a sparse version of diag(1:5)new("matrix.csr") # the 1x1 matrix [0] new("matrix.diag.csr") # the 'same' as(1:5, "matrix.diag.csr") # a sparse version of diag(1:5)
A class of objects returned from Ng and Peyton's (1993) block sparse Cholesky algorithm.
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.
Objects may be created by calls of the form new("matrix.csr.chol",
...), but typically result from
chol(<matrix.csr>).
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.
signature(x = "matrix.csr.chol",
upper.tri=TRUE): to get the sparse ("matrix.csr") upper
triangular matrix corresponding to the Cholesky decomposition.
signature(r = "matrix.csr.chol"): for computing
when the Cholesky decomposition is .
Base R's chol and SparseM's
chol, notably for examples;
backsolve
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)$values # all positive (but close to singular) ch5g <- chol(x5g) str(ch5g) # --> the slots of the "matrix.csr.chol" class mch5g <- as.matrix.csr(ch5g) print.table(as.matrix(mch5g), zero.print=".") # indeed upper triagonal w/ positive diagonal ## x5 has even more extreme entry at [5,5]: x5 <- x5g; x5[5,5] <- 2.9e32 m5 <- as.matrix(x5) (c5 <- chol(m5))# still fine, w/ [5,5] entry = 1.7e16 and other diag.entries in (9.56, 17.32) ch5 <- chol(x5) # --> warning "Replaced 3 tiny diagonal entries by 'Large'" # gave error for a while (mmc5 <- as.matrix(as.matrix.csr(ch5))) # yes, these replacements were extreme, and the result is "strange' ## Solve the problem (here) specifying non-default singularity-tuning par 'tiny': ch5. <- chol(x5, tiny = 1e-33) (mmc5. <- as.matrix(as.matrix.csr(ch5.))) # looks much better. ## Indeed: R'R back-permuted *is* the original matrix x5, here m5: (RtR <- crossprod(mmc5.)[ch5.@invp, ch5.@invp]) all.equal(m5, RtR, tolerance = 2^-52) stopifnot(all.equal(m5, RtR, tolerance = 1e-14)) # on F38 Linux, only need tol = 1.25e-16x5g <- new("matrix.csr", ra = c(300, 130, 5, 130, 330, 125, 10, 5, 125, 200, 70, 10, 70, 121.5, 1e30), ja = c(1:3, 1:4, 1:4, 2:5), ia = c(1L, 4L, 8L, 12L, 15L, 16L), dimension = c(5L, 5L)) (m5g <- as.matrix(x5g)) # yes, is symmetric, and positive definite: eigen(m5g, only.values=TRUE)$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
A class for sparse matrices stored in symmetric sparse column ('ssc') format.
Objects can be created by calls of the form new("matrix.ssc", ...).
ra:Object of class numeric, a real array of nnz elements containing the non-zero
elements of the lower triangular part of A, stored in column order. Thus, if i<j, all elements
of column i precede elements from column j. The order of elements
within the column is immaterial.
ja:Object of class integer, an integer array of nnz elements containing the row
indices of the elements stored in ‘ra’.
ia:Object of class integer, an integer array of n+1 elements containing pointers to
the beginning of each column in the arrays ‘ra’ and ‘ja’. Thus
‘ia[i]’ indicates the position in the arrays ‘ra’ and ‘ja’
where the ith column begins. The last, (n+1)st, element of ‘ia’
indicates where the n+1 column would start, if it existed.
dimension:Object of class integer, dimension of the matrix
signature(x = "matrix.ssc"): ...
signature(x = "matrix.ssc"): ...
signature(x = "matrix.ssc"): ...
signature(x = "matrix.ssc"): ...
signature(x = "matrix.ssc"): ...
A new class consists of the coefficient matrix and the right-hand-side of a linear system of equations, initial guess of the solution and the exact solutions if they exist stored in external files using the Harwell-Boeing format.
Objects can be created by calls of the form new("matrix.ssc.hb", ...).
ra:Object of class numeric, ra component of the csc or ssc format of the coefficient matrix, X.
ja:Object of class integer, ja component of the csc or s
sc format of the coefficient matrix, X.
ia:Object of class integer, ia component of the csc or ssc format of the coefficient matrix, X.
rhs.ra:Object of class numeric, ra component of the right-hand-side, y, if stored in csc or
ssc format; right-hand-side stored in dense vector or matrix otherwise.
guess:Object of class numeric or NULL vector of the initial guess of the solutions if they exist;
a null vector otherwise.
xexact:Object of class numeric or NULL vector of the exact solutions, b, if they exist; a null vector otherwise.
dimension:Object of class integer, dimenson of the coefficient matrix, X.
rhs.dim:Object of class integer, dimenson of the right-hand-side, y.
rhs.mode:Object of class character or NULL storage mode of the right-hand-side; can be full storage or
same format as the coefficient matrix.
Class "matrix.csc.hb", directly.
signature(object = "matrix.ssc.hb"): ...
model.matrix, model.response,
read.matrix.hb, matrix.csc.hb-class
A class for sparse matrices stored in symmetric sparse row ('ssr') format.
Objects can be created by calls of the form new("matrix.ssr", ...).
ra:Object of class numeric, a real array of nnz elements containing the non-zero
elements of the lower triangular part of A, stored in row order. Thus, if i<j, all elements
of row i precede elements from row j. The order of elements
within the rows is immaterial.
ja:Object of class integer, an integer array of nnz elements containing the column
indices of the elements stored in ‘ra’.
ia:Object of class integer, an integer array of n+1 elements containing pointers to
the beginning of each row in the arrays ‘ra’ and ‘ja’. Thus
‘ia[i]’ indicates the position in the arrays ‘ra’ and ‘ja’
where the ith row begins. The last, (n+1)st, element of ‘ia’
indicates where the n+1 row would start, if it existed.
dimension:Object of class integer, dimension of the matrix
signature(x = "matrix.ssr"): ...
signature(x = "matrix.ssr"): ...
signature(x = "matrix.ssr"): ...
signature(x = "matrix.ssr"): ...
signature(x = "matrix.ssr"): ...
ssr <- as.matrix.ssr(diag(c(2,3,5,7))) ssrssr <- as.matrix.ssr(diag(c(2,3,5,7))) ssr
A sparse extension of lm
Objects can be created by calls of the form new("mslm", ...).
coefficients:Object of class numeric estimated coefficients
chol:Object of class matrix.csr.chol generated by
the function chol
residuals:Object of class "numeric" residuals
fitted:Object of class "numeric" fitted values
Class "lm", directly.
Class "slm", directly.
Class "oldClass", by class "lm".
signature(object = "mslm"): ...
signature(object = "mslm"): ...
signature(object = "mslm"): ...
signature(object = "mslm"): ...
A virtual class needed by the "matrix.csc.hb" class
A virtual Class: No objects may be created from it.
No methods defined with class "numeric or NULL" in the signature.
This is a function to illustrate the use of sparse linear algebra
to solve a linear least squares problem using Cholesky decomposition.
The syntax and output attempt to emulate lm() but may
fail to do so fully satisfactorily. Ideally, this would eventually
become a method for lm. The main obstacle to this step is
that it would be necessary to have a model.matrix function that
returned an object in sparse csr form. For the present, the objects
represented in the formula must be in dense form. If the user wishes
to specify fitting with a design matrix that is already in sparse form,
then the lower level function slm.fit() should be used.
slm(formula, data, weights, na.action, method = "csr", contrasts = NULL, ...)slm(formula, data, weights, na.action, method = "csr", contrasts = NULL, ...)
formula |
a formula object, with the response on the left of a |
data |
a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument. If this is missing, then the variables in the formula should be on the search list. This may also be a single number to handle some special cases – see below for details. |
weights |
vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous. |
na.action |
a function to filter missing data.
This is applied to the model.frame after any subset argument has been used.
The default (with |
method |
there is only one method based on Cholesky factorization |
contrasts |
a list giving contrasts for some or all of the factors
default = |
... |
additional arguments for the fitting routines |
A list of class slm consisting of:
coefficients |
estimated coefficients |
chol |
cholesky object from fitting |
residuals |
residuals |
fitted |
fitted values |
terms |
terms |
call |
call |
...
Roger Koenker
Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
slm.methods for methods summary, print, fitted,
residuals and coef associated with class slm,
and slm.fit for lower level fitting functions. The latter functions
are of special interest if you would like to pass a sparse form of the
design matrix directly to the fitting process.
data(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs X1 <- as.matrix(X) slm.time <- system.time(slm(y~X1-1) -> slm.o) # pretty fast lm.time <- system.time(lm(y~X1-1) -> lm.o) # very slow cat("slm time =",slm.time,"\n") cat("slm Results: Reported Coefficients Truncated to 5 ","\n") sum.slm <- summary(slm.o) sum.slm$coef <- sum.slm$coef[1:5,] sum.slm cat("lm time =",lm.time,"\n") cat("lm Results: Reported Coefficients Truncated to 5 ","\n") sum.lm <- summary(lm.o) sum.lm$coef <- sum.lm$coef[1:5,] sum.lmdata(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs X1 <- as.matrix(X) slm.time <- system.time(slm(y~X1-1) -> slm.o) # pretty fast lm.time <- system.time(lm(y~X1-1) -> lm.o) # very slow cat("slm time =",slm.time,"\n") cat("slm Results: Reported Coefficients Truncated to 5 ","\n") sum.slm <- summary(slm.o) sum.slm$coef <- sum.slm$coef[1:5,] sum.slm cat("lm time =",lm.time,"\n") cat("lm Results: Reported Coefficients Truncated to 5 ","\n") sum.lm <- summary(lm.o) sum.lm$coef <- sum.lm$coef[1:5,] sum.lm
A sparse extension of lm
Objects can be created by calls of the form new("slm", ...).
coefficients:Object of class numeric estimated coefficients
chol:Object of class matrix.csr.chol generated by
function chol
residuals:Object of class "numeric" residuals
fitted:Object of class "numeric" fitted values
Class "lm", directly.
Class "oldClass", by class "lm".
signature(object = "slm"): ...
signature(object = "slm"): ...
signature(object = "slm"): ...
signature(object = "slm"): ...
Fitting functions for sparse linear model fitting.
slm.fit(x,y,method, ...) slm.wfit(x,y,weights,...) slm.fit.csr(x, y, ...)slm.fit(x,y,method, ...) slm.wfit(x,y,weights,...) slm.fit.csr(x, y, ...)
x |
design matrix. |
y |
vector of response observations. |
method |
only |
weights |
an optional vector of weights to be used in the fitting process. If specified, weighted least squares is used with weights ‘weights’ (that is, minimizing
The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous. |
... |
additional arguments. |
slm.fit and slm.wfit call slm.fit.csr
to do Cholesky decomposition
and then backsolve to obtain the least squares estimated coefficients.
These functions can be called directly if the user is willing to
specify the design matrix in matrix.csr form. This is often
advantageous in large problems to reduce memory requirements.
A list of class slm consisting of:
coef |
estimated coefficients |
chol |
cholesky object from fitting |
residuals |
residuals |
fitted |
fitted values |
df.residual |
degrees of freedom |
terms |
terms |
call |
call |
...
Roger Koenker
Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
data(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs class(X) # -> "matrix.csr" class(y) # -> NULL slm.fit(X,y)->slm.fit.o # this is much more efficient in memory usage than slm() slm(y~as.matrix(X)-1) -> slm.o # this requires X to be transformed into dense mode cat("Difference between `slm.fit' and `slm' estimated coefficients =", sum(abs(slm.fit.o$coef-slm.o$coef)),"\n")data(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs class(X) # -> "matrix.csr" class(y) # -> NULL slm.fit(X,y)->slm.fit.o # this is much more efficient in memory usage than slm() slm(y~as.matrix(X)-1) -> slm.o # this requires X to be transformed into dense mode cat("Difference between `slm.fit' and `slm' estimated coefficients =", sum(abs(slm.fit.o$coef-slm.o$coef)),"\n")
Summarize, print, and extract objects from slm objects.
## S3 method for class 'slm' summary(object, correlation, ...) ## S3 method for class 'mslm' summary(object, ...) ## S3 method for class 'slm' print(x, digits, ...) ## S3 method for class 'summary.slm' print(x, digits, symbolic.cor, signif.stars, ...) ## S3 method for class 'slm' fitted(object, ...) ## S3 method for class 'slm' residuals(object, ...) ## S3 method for class 'slm' coef(object, ...) ## S3 method for class 'slm' extractAIC(fit, scale = 0, k = 2, ...) ## S3 method for class 'slm' deviance(object, ...)## S3 method for class 'slm' summary(object, correlation, ...) ## S3 method for class 'mslm' summary(object, ...) ## S3 method for class 'slm' print(x, digits, ...) ## S3 method for class 'summary.slm' print(x, digits, symbolic.cor, signif.stars, ...) ## S3 method for class 'slm' fitted(object, ...) ## S3 method for class 'slm' residuals(object, ...) ## S3 method for class 'slm' coef(object, ...) ## S3 method for class 'slm' extractAIC(fit, scale = 0, k = 2, ...) ## S3 method for class 'slm' deviance(object, ...)
object, x, fit
|
object of class |
digits |
minimum number of significant digits to be used for most numbers. |
scale |
optional numeric specifying the scale parameter of the model, see 'scale' in 'step'. Currently only used in the '"lm"' method, where 'scale' specifies the estimate of the error variance, and 'scale = 0' indicates that it is to be estimated by maximum likelihood. |
k |
numeric specifying the "weight" of the equivalent degrees of freedom ('edf') part in the AIC formula. |
symbolic.cor |
logical; if |
signif.stars |
logical; if |
correlation |
logical; if |
... |
additional arguments passed to methods. |
print.slm and print.summary.slm return invisibly.
fitted.slm, residuals.slm, and coef.slm
return the corresponding components of the slm object.
extractAIC.slm and deviance.slm return the AIC
and deviance values of the fitted object.
Roger Koenker
Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
slm
data(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs X1 <- as.matrix(X) slm.time <- system.time(slm(y~X1-1) -> slm.o) # pretty fast cat("slm time =",slm.time,"\n") cat("slm Results: Reported Coefficients Truncated to 5 ","\n") sum.slm <- summary(slm.o) sum.slm$coef <- sum.slm$coef[1:5,] sum.slm fitted(slm.o)[1:10] residuals(slm.o)[1:10] coef(slm.o)[1:10]data(lsq) X <- model.matrix(lsq) #extract the design matrix y <- model.response(lsq) # extract the rhs X1 <- as.matrix(X) slm.time <- system.time(slm(y~X1-1) -> slm.o) # pretty fast cat("slm time =",slm.time,"\n") cat("slm Results: Reported Coefficients Truncated to 5 ","\n") sum.slm <- summary(slm.o) sum.slm$coef <- sum.slm$coef[1:5,] sum.slm fitted(slm.o)[1:10] residuals(slm.o)[1:10] coef(slm.o)[1:10]
Read, and extract components of data in Harwell-Boeing sparse matrix format.
read.matrix.hb(file) model.matrix(object, ...) model.response(data,type)read.matrix.hb(file) model.matrix(object, ...) model.response(data,type)
file |
file name to read from or |
data, object
|
an object of either 'matrix.csc.hb' or 'matrix.ssc.hb' class |
type |
One of ‘"any"’, ‘"numeric"’, ‘"double"’. Using the either of latter two coerces the result to have storage mode ‘"double"’ |
... |
additional arguments to model.matrix |
Sparse coefficient matrices in the Harwell-Boeing format are stored in
80-column records. Each file begins with a multiple line header block
followed by two, three or four data blocks. The header block contains
summary information on the storage formats and storage requirements.
The data blocks contain information of the sparse coefficient matrix and
data for the right-hand-side of the linear system of equations,
initial guess of the solution and the exact solutions if they exist.
The function model.matrix extracts the X matrix component.
The function model.response extracts the y vector (or matrix).
The function model.guess extracts the guess vector.
The function model.xexact extracts the xexact vector.
This function is written in R replacing a prior implementation based
on iohb.c which had memory fault difficulties. The function write.matrix.hb
has been purged; users wishing to write matrices in Harwell-Boeing format
are advised to convert SparseM matrices to Matrix classes and use writeHB
from the Matrix package. Contributions of code to facilitate this conversion
would be appreciated!
The function read.matrix.hb returns a list of class
matrix.csc.hb or matrix.ssc.hb depending
on how the coefficient matrix is stored in the file.
ra |
ra component of the csc or ssc format of the coefficient matrix, X. |
ja |
ja component of the csc or ssc format of the coefficient matrix, X. |
ia |
ia component of the csc or ssc format of the coefficient matrix, X. |
rhs.ra |
ra component of the right-hand-side, y, if stored in csc or ssc format; right-hand-side stored in dense vector or matrix otherwise. |
rhs.ja |
ja component of the right-hand-side, y, if stored in csc or ssc format; a null vector otherwise. |
rhs.ia |
ia component of the right-hand-side, y, if stored in csc or ssc format; a null vector otherwise. |
xexact |
vector of the exact solutions, b, if they exist; a null vector otherwise. |
guess |
vector of the initial guess of the solutions if they exist; a null vector otherwise. |
dimension |
dimenson of the coefficient matrix, X. |
rhs.dim |
dimenson of the right-hand-side, y. |
rhs.mode |
storage mode of the right-hand-side; can be full storage or same format as the coefficient matrix, for the moment the only allowed mode is "F" for full, or dense mode. |
The function model.matrix returns the X matrix of class matrix.csr.
The function model.response returns the y vector (or matrix).
The function model.guess returns the guess vector (or matrix).
The function model.xexact returns the xexact vector (or matrix).
Pin Ng
Duff, I.S., Grimes, R.G. and Lewis, J.G. (1992) User's Guide for Harwell-Boeing Sparse Matrix Collection at https://math.nist.gov/MatrixMarket/collections/hb.html
slm for sparse version of lm SparseM.ops for operators on class matrix.csr SparseM.solve for linear equation solving for class matrix.csr SparseM.image for image plotting of class matrix.csr SparseM.ontology for coercion of class matrix.csr
Xy <- read.matrix.hb(system.file("extdata","lsq.rra",package = "SparseM")) class(Xy) # -> [1] "matrix.csc.hb" X <- model.matrix(Xy)->X class(X) # -> "matrix.csr" dim(X) # -> [1] 1850 712 y <- model.response(Xy) # extract the rhs length(y) # [1] 1850 Xy <- read.matrix.hb(system.file("extdata","rua_32_ax.rua",package = "SparseM")) X <- model.matrix(Xy) y <- model.response(Xy) # extract the rhs g <- model.guess(Xy) # extract the guess a <- model.xexact(Xy) # extract the xexact fit <- solve(t(X) %*% X, t(X) %*% y) # compare solution with xexact solutionXy <- read.matrix.hb(system.file("extdata","lsq.rra",package = "SparseM")) class(Xy) # -> [1] "matrix.csc.hb" X <- model.matrix(Xy)->X class(X) # -> "matrix.csr" dim(X) # -> [1] 1850 712 y <- model.response(Xy) # extract the rhs length(y) # [1] 1850 Xy <- read.matrix.hb(system.file("extdata","rua_32_ax.rua",package = "SparseM")) X <- model.matrix(Xy) y <- model.response(Xy) # extract the rhs g <- model.guess(Xy) # extract the guess a <- model.xexact(Xy) # extract the xexact fit <- solve(t(X) %*% X, t(X) %*% y) # compare solution with xexact solution
Display the pattern of non-zero entries of
a matrix of class matrix.csr.
## S4 method for signature 'matrix.csr' image(x, col=c("white","gray"), xlab="column", ylab="row", ...)## S4 method for signature 'matrix.csr' image(x, col=c("white","gray"), xlab="column", ylab="row", ...)
x |
a matrix of class |
col |
a list of colors such as that generated by |
xlab, ylab
|
each a character string giving the labels for the x and y axis. |
... |
additional arguments. |
The pattern of the non-zero entries of a sparse matrix is displayed. By default nonzero entries of the matrix appear as gray blocks and zero entries as white background.
Koenker, R and Ng, P. (2002).
SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
SparseM.ops,
SparseM.solve,
SparseM.ontology
a <- rnorm(20*5) A <- matrix(a,20,5) A[row(A)>col(A)+4|row(A)<col(A)+3] <- 0 b <- rnorm(20*5) B <- matrix(b,20,5) B[row(A)>col(A)+2|row(A)<col(A)+2] <- 0 image(as.matrix.csr(A)%*%as.matrix.csr(t(B)))a <- rnorm(20*5) A <- matrix(a,20,5) A[row(A)>col(A)+4|row(A)<col(A)+3] <- 0 b <- rnorm(20*5) B <- matrix(b,20,5) B[row(A)>col(A)+2|row(A)<col(A)+2] <- 0 image(as.matrix.csr(A)%*%as.matrix.csr(t(B)))
This group of functions evaluates and coerces changes in class structure.
as.matrix.csr(x, nrow, ncol, eps = .Machine$double.eps, ...) ## S4 method for signature 'matrix.csr.chol' as.matrix.csr(x, nrow, ncol, eps, upper.tri=TRUE, ...) ## S4 method for signature 'matrix.csr' as.matrix.csc(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.coo' as.matrix.ssr(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.csc' as.matrix.ssc(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.csr' as.matrix.coo(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) is.matrix.csr(x) is.matrix.csc(x) is.matrix.ssr(x) is.matrix.ssc(x) is.matrix.coo(x)as.matrix.csr(x, nrow, ncol, eps = .Machine$double.eps, ...) ## S4 method for signature 'matrix.csr.chol' as.matrix.csr(x, nrow, ncol, eps, upper.tri=TRUE, ...) ## S4 method for signature 'matrix.csr' as.matrix.csc(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.coo' as.matrix.ssr(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.csc' as.matrix.ssc(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) ## S4 method for signature 'matrix.csr' as.matrix.coo(x, nrow = 1, ncol = 1, eps = .Machine$double.eps) is.matrix.csr(x) is.matrix.csc(x) is.matrix.ssr(x) is.matrix.ssc(x) is.matrix.coo(x)
x |
is a matrix, or vector object, of either dense or sparse form |
nrow |
number of rows of matrix |
ncol |
number of columns of matrix |
eps |
A tolerance parameter: elements of x such that abs(x) < eps set to zero.
This argument is only relevant when coercing matrices from dense to sparse form. Defaults to
|
upper.tri |
|
... |
other arguments |
The function matrix.csc acts like matrix to coerce a vector object to
a sparse matrix object of class matrix.csr.
This aspect of the code is in the process of conversion from S3 to S4 classes.
For the most part the S3 syntax prevails. An exception is the code to
coerce vectors to diagonal matrix form which uses as(v,"matrix.diag.csr".
The generic functions as.matrix.xxx coerce a matrix x into
a matrix of storage class matrix.xxx. The argument matrix x
may be of conventional dense form, or of any of the four supported
classes: matrix.csr, matrix.csc, matrix.ssr, matrix.ssc.
The generic functions is.matrix.xxx evaluate whether the
argument is of class matrix.xxx. The function
as.matrix transforms a matrix of any sparse class into conventional
dense form. The primary storage class for sparse matrices is the
compressed sparse row matrix.csr class.
An n by m matrix A with real elements ,
stored in matrix.csr format consists of three arrays:
ra: a real array of nnz elements containing the non-zero
elements of A, stored in row order. Thus, if i<j, all elements of row i
precede elements from row j. The order of elements within the rows is immaterial.
ja: an integer array of nnz elements containing the column
indices of the elements stored in ra.
ia: an integer array of n+1 elements containing pointers to
the beginning of each row in the arrays ra and ja. Thus
ia[i] indicates the position in the arrays ra and
ja where the ith row begins. The last, (n+1)st, element of
ia indicates where the n+1 row would start, if it existed.
The compressed sparse column class matrix.csc is defined in
an analogous way, as are the matrix.ssr, symmetric sparse row, and
matrix.ssc, symmetric sparse column classes.
as.matrix.ssr and as.matrix.ssc should ONLY be used with
symmetric matrices.
as.matrix.csr(x), when x is an object of class matrix.csr.chol
(that is, an object returned by a call to chol(a) when a
is an object of class matrix.csr or matric.csc),
by default returns an upper triangular matrix, which
is not consistent with the result of chol in the base
package. To get an lower triangular matric.csr matrix, use either
as.matrix.csr(x, upper.tri = FALSE) or
t(as.matrix.csr(x)).
Koenker, R and Ng, P. (2002) SparseM: A Sparse Matrix Package for R. http://www.econ.uiuc.edu/~roger/research/home.html
SparseM.hb for handling Harwell-Boeing sparse matrices.
t(m5 <- as.matrix.csr(c(-1:1,0,0))) t(M4 <- as.matrix.csc(c(0:2,0), 4)) (S3 <- as.matrix.ssr(diag(x = 0:2))) # *symmetric* stopifnot(identical(dim(m5), c(5L, 1L)), identical(dim(M4), c(4L, 1L)), identical(dim(S3), c(3L, 3L))) n1 <- 10 p <- 5 a <- round(rnorm(n1*p), 2) a[abs(a) < 0.7] <- 0 A <- matrix(a,n1,p) B <- t(A) %*% A A.csr <- as.matrix.csr(A) A.csc <- as.matrix.csc(A) B.ssr <- as.matrix.ssr(B) B.ssc <- as.matrix.ssc(B) stopifnot(exprs = { is.matrix.csr(A.csr) # -> TRUE is.matrix.csc(A.csc) # -> TRUE is.matrix.ssr(B.ssr) # -> TRUE is.matrix.ssc(B.ssc) # -> TRUE }) as.matrix(A.csr) as.matrix(A.csc) as.matrix(B.ssr) as.matrix(B.ssc) as.matrix.csr(0, 2,3) # sparse matrix of all zeros ## Diagonal (sparse) : as(4, "matrix.diag.csr") # identity matrix of dimension 4 as(2:0, "matrix.diag.csr") # diagonal 3x3 matrixt(m5 <- as.matrix.csr(c(-1:1,0,0))) t(M4 <- as.matrix.csc(c(0:2,0), 4)) (S3 <- as.matrix.ssr(diag(x = 0:2))) # *symmetric* stopifnot(identical(dim(m5), c(5L, 1L)), identical(dim(M4), c(4L, 1L)), identical(dim(S3), c(3L, 3L))) n1 <- 10 p <- 5 a <- round(rnorm(n1*p), 2) a[abs(a) < 0.7] <- 0 A <- matrix(a,n1,p) B <- t(A) %*% A A.csr <- as.matrix.csr(A) A.csc <- as.matrix.csc(A) B.ssr <- as.matrix.ssr(B) B.ssc <- as.matrix.ssc(B) stopifnot(exprs = { is.matrix.csr(A.csr) # -> TRUE is.matrix.csc(A.csc) # -> TRUE is.matrix.ssr(B.ssr) # -> TRUE is.matrix.ssc(B.ssc) # -> TRUE }) as.matrix(A.csr) as.matrix(A.csc) as.matrix(B.ssr) as.matrix(B.ssc) as.matrix.csr(0, 2,3) # sparse matrix of all zeros ## Diagonal (sparse) : as(4, "matrix.diag.csr") # identity matrix of dimension 4 as(2:0, "matrix.diag.csr") # diagonal 3x3 matrix
Basic linear algebra operations for sparse matrices,
mostly of class matrix.csr.
x |
matrix of class |
y |
matrix of class |
value |
replacement values. |
i, j
|
vectors of elements to extract or replace. |
nrow |
optional number of rows for the result. |
lag |
an integer indicating which lag to use. |
differences |
an integer indicating the order of the difference. |
Linear algebra operations for matrices of class
matrix.csr are designed to behave exactly as for
regular matrices. In particular, matrix multiplication, kronecker
product, addition,
subtraction and various logical operations should work as with the conventional
dense form of matrix storage, as does indexing, rbind, cbind, and diagonal
assignment and extraction. The method diag may be used to extract the
diagonal of a matrix.csr object, to create a sparse diagonal see
SparseM.ontology.
The function determinant computes the (log) determinant,
of the argument, returning a "det" object as the base function.
This is typically preferred over using the function det()
which is a simple wrapper for determinant(), in a way it will work
for our sparse matrices, as well.
determinant() computes the determinant of the argument
matrix. If the matrix is of class matrix.csr then it must
be symmetric, or an error will be returned. If the matrix is of
class matrix.csr.chol then the (pre-computed) determinant of the Cholesky
factor is returned, i.e., the product of the diagonal elements.
The function norm, i.e. norm(x, type), by default computes
the “sup” (or "M"aximum norm, i.e., the maximum of the matrix
elements. Optionally, this type = "sup" (equivalently, type = "M") norm can
be replaced by the Hilbert-Schmidt, type = "HS" or equivalently, type = "F"
norm, or the type = "l1", norm.
Note that for historical reasons, the default type differs from R's
own norm(), see the examples using B, below.
The "sup" === "M" and "HS" === "F" equivalences have been
introduced in SparseM version 1.84.
Koenker, R and Ng, P. (2002).
SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html
slm for sparse linear model fitting.
SparseM.ontology for coercion and other class relations
involving our sparse matrix classes.
n1 <- 10 n2 <- 10 p <- 6 y <- rnorm(n1) a <- round(rnorm(n1*p), 2) a[abs(a) < 0.5] <- 0 A <- matrix(a, n1,p) A.csr <- as.matrix.csr(A) B <- matrix(c(1.5, 0, 0, 0, -1.4, 0, 0, 0, 0, -1.4, 2, 0, -1, 0, 0, 2.1, -1.9, 1.4, 0, 0, 0,-2.3, 0, 0, -1.9, 0, 0, 0, 0, -1.4, 0, 0, 0, 0, 0, -3, 0, 1.3, 0, 1.1, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -1.6,0, 0, 0, 0, 0, -1.7,0), 10L, 6L) rcond(B) # 0.21 .. i.e., quite well conditioned B.csr <- as.matrix.csr(B) B.csr ## norm() : different 'type' for base R and SparseM: (nR <- vapply(c("1", "I", "F", "M", "2"), norm, 0, x = B)) ## 1 I F M 2 ## 8.400000 5.300000 7.372923 3.000000 4.464801 (nSpM <- vapply(c("sup","M", "HS","F", "l1"), norm, 0, x = B.csr)) ## sup M HS F l1 ## 3.000000 3.000000 7.372923 7.372923 30.000000 stopifnot(all.equal(unname(nSpM[c("M", "F")]), unname(nR [c("M", "F" )]), tolerance = 1e-14)) # matrix transposition and multiplication BtB <- crossprod(B) # == t(B) %*% B {more efficiently} A.csr %*% t(B.csr) BtBs <- t(B.csr) %*% B.csr BtBs stopifnot(all.equal( BtB, as.matrix(BtBs), tolerance = 1e-14), all.equal(det(BtB), print(det(BtBs)), tolerance = 1e-14)) # matrix o vector stopifnot(all.equal(y %*% A , y %*% A.csr) , all.equal(A %*% 1:6, A.csr %*% 1:6) ) # kronecker product - via kronecker() methods: A.csr %x% matrix(1:4,2,2)n1 <- 10 n2 <- 10 p <- 6 y <- rnorm(n1) a <- round(rnorm(n1*p), 2) a[abs(a) < 0.5] <- 0 A <- matrix(a, n1,p) A.csr <- as.matrix.csr(A) B <- matrix(c(1.5, 0, 0, 0, -1.4, 0, 0, 0, 0, -1.4, 2, 0, -1, 0, 0, 2.1, -1.9, 1.4, 0, 0, 0,-2.3, 0, 0, -1.9, 0, 0, 0, 0, -1.4, 0, 0, 0, 0, 0, -3, 0, 1.3, 0, 1.1, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -1.6,0, 0, 0, 0, 0, -1.7,0), 10L, 6L) rcond(B) # 0.21 .. i.e., quite well conditioned B.csr <- as.matrix.csr(B) B.csr ## norm() : different 'type' for base R and SparseM: (nR <- vapply(c("1", "I", "F", "M", "2"), norm, 0, x = B)) ## 1 I F M 2 ## 8.400000 5.300000 7.372923 3.000000 4.464801 (nSpM <- vapply(c("sup","M", "HS","F", "l1"), norm, 0, x = B.csr)) ## sup M HS F l1 ## 3.000000 3.000000 7.372923 7.372923 30.000000 stopifnot(all.equal(unname(nSpM[c("M", "F")]), unname(nR [c("M", "F" )]), tolerance = 1e-14)) # matrix transposition and multiplication BtB <- crossprod(B) # == t(B) %*% B {more efficiently} A.csr %*% t(B.csr) BtBs <- t(B.csr) %*% B.csr BtBs stopifnot(all.equal( BtB, as.matrix(BtBs), tolerance = 1e-14), all.equal(det(BtB), print(det(BtBs)), tolerance = 1e-14)) # matrix o vector stopifnot(all.equal(y %*% A , y %*% A.csr) , all.equal(A %*% 1:6, A.csr %*% 1:6) ) # kronecker product - via kronecker() methods: A.csr %x% matrix(1:4,2,2)
chol()performs a Cholesky decomposition of a symmetric
positive definite sparse matrix x of class matrix.csr.
backsolve()performs a triangular back-fitting to compute the solutions of a system of linear equations in one step.
backsolve() and forwardsolve()
can also split the functionality of
backsolve into two steps.
solve()combines chol() and backsolve() to
compute the inverse of a matrix if the right-hand-side is missing.
chol(x, ...) ## S4 method for signature 'matrix.csr' chol(x, pivot = FALSE, nsubmax, nnzlmax, tmpmax, eps = .Machine$double.eps, tiny = 1e-30, Large = 1e128, warnOnly = FALSE, cacheKb = 1024L, level = 8L, ...) ## S4 method for signature 'matrix.csr.chol' backsolve(r, x, k, upper.tri, transpose, twice = TRUE, drop = TRUE, ...) ## S4 method for signature 'matrix.csr.chol' forwardsolve(l, x, k, upper.tri, transpose) ## S4 method for signature 'matrix.csr' solve(a, b, ...)chol(x, ...) ## S4 method for signature 'matrix.csr' chol(x, pivot = FALSE, nsubmax, nnzlmax, tmpmax, eps = .Machine$double.eps, tiny = 1e-30, Large = 1e128, warnOnly = FALSE, cacheKb = 1024L, level = 8L, ...) ## S4 method for signature 'matrix.csr.chol' backsolve(r, x, k, upper.tri, transpose, twice = TRUE, drop = TRUE, ...) ## S4 method for signature 'matrix.csr.chol' forwardsolve(l, x, k, upper.tri, transpose) ## S4 method for signature 'matrix.csr' solve(a, b, ...)
a |
symmetric positive definite matrix of class |
r, l
|
object of class |
x |
|
b |
vector or matrix right-hand-side(s) to solve for. |
k |
inherited from the generic; not used here. |
pivot |
inherited from the generic; not used here. |
nsubmax, nnzlmax, tmpmax
|
positive integer numbers with smart defaults; do not set unless you know what you are doing! |
eps |
positive tolerance for checking symmetry; change with caution. |
tiny |
positive tolerance for checking diagonal entries to be
“essentially zero” and hence to be replaced by |
Large |
large positive number, “essentially infinite”, to replace tiny diagonal entries during Cholesky. |
warnOnly |
|
cacheKb |
a positive integer, specifying an approximate size of the machine's cache memory in kilo (1024) bytes (‘Kb’); used to be hard wired to 64. |
level |
level of loop unrolling while performing numerical
factorization; an integer in |
upper.tri, transpose
|
inherited from the generic; not used here. |
twice |
|
drop |
|
... |
further arguments passed to or from other methods. |
chol performs a Cholesky decomposition of
a symmetric positive definite sparse matrix a of class
matrix.csr using the block sparse Cholesky algorithm of Ng and
Peyton (1993). The structure of the resulting matrix.csr.chol
object is relatively complicated. If necessary it can be coerced back
to a matrix.csr object as usual with as.matrix.csr.
backsolve does triangular back-fitting to compute
the solutions of a system of linear equations. For systems of linear equations
that only vary on the right-hand-side, the result from chol
can be reused. Contrary to the behavior of backsolve in base R,
the default behavior of backsolve(C,b) when C is a matrix.csr.chol object
is to produce a solution to the system where C <- chol(A), see
the example section. When the flag twice is FALSE then backsolve
solves the system , up to a permutation – see the comments below.
The command solve combines chol and backsolve, and will
compute the inverse of a matrix if the right-hand-side is missing.
The determinant of the Cholesky factor is returned providing a
means to efficiently compute the determinant of sparse positive
definite symmetric matrices.
There are several integer storage parameters that are set by default in the call
to the Cholesky factorization, these can be overridden in any of the above
functions and will be passed by the usual "dots" mechanism. The necessity
to do this is usually apparent from error messages like: Error
in local(X...) increase tmpmax. For example, one can use,
solve(A,b, tmpmax = 100*nrow(A)). The current default for tmpmax
is 50*nrow(A). Some experimentation may be needed to
select appropriate values, since they are highly problem dependent. See
the code of chol() for further details on the current defaults.
There is no explicit checking for positive definiteness of the matrix
so users are advised to ensure that this condition is satisfied.
Messages such as "insufficient space" may indicate that one is trying
to factor a singular matrix.
Because the sparse Cholesky algorithm re-orders the positive
definite sparse matrix A, the value of
x <- backsolve(C, b) does not equal the solution to the
triangular system , but is instead the solution to the
system for some permutation matrix
(and analogously for x <- forwardsolve(C, b)). However, a
little algebra easily shows that
backsolve(C, forwardsolve(C, b), twice = FALSE) is the solution
to the equation . Finally, if C <- chol(A) for some
sparse covariance matrix A, and z is a conformable standard normal vector,
then the product y <- as.matrix.csr(C) %*% z is normal with covariance
matrix A irrespective of the permutation of the Cholesky factor.
Koenker, R and Ng, P. (2002) SparseM: A Sparse Matrix Package for R. http://www.econ.uiuc.edu/~roger/research/home.html
Ng, E. G. and B. W. Peyton (1993) Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J. Scientific Computing 14, 1034–1056.
slm() for a sparse version of stats package's lm().
data(lsq) class(lsq) # -> [1] "matrix.csc.hb" model.matrix(lsq)->design.o class(design.o) # -> "matrix.csr" dim(design.o) # -> [1] 1850 712 y <- model.response(lsq) # extract the rhs length(y) # [1] 1850 X <- as.matrix(design.o) c(object.size(X) / object.size(design.o)) ## X is 92.7 times larger t(design.o) %*% design.o -> XpX t(design.o) %*% y -> Xpy chol(XpX) -> chol.o determinant(chol.o) b1 <- backsolve(chol.o,Xpy) # least squares solutions in two steps b2 <- solve(XpX,Xpy) # least squares estimates in one step b3 <- backsolve(chol.o, forwardsolve(chol.o, Xpy), twice = FALSE) # in three steps ## checking that these three are indeed equal : stopifnot(all.equal(b1, b2), all.equal(b2, b3))data(lsq) class(lsq) # -> [1] "matrix.csc.hb" model.matrix(lsq)->design.o class(design.o) # -> "matrix.csr" dim(design.o) # -> [1] 1850 712 y <- model.response(lsq) # extract the rhs length(y) # [1] 1850 X <- as.matrix(design.o) c(object.size(X) / object.size(design.o)) ## X is 92.7 times larger t(design.o) %*% design.o -> XpX t(design.o) %*% y -> Xpy chol(XpX) -> chol.o determinant(chol.o) b1 <- backsolve(chol.o,Xpy) # least squares solutions in two steps b2 <- solve(XpX,Xpy) # least squares estimates in one step b3 <- backsolve(chol.o, forwardsolve(chol.o, Xpy), twice = FALSE) # in three steps ## checking that these three are indeed equal : stopifnot(all.equal(b1, b2), all.equal(b2, b3))
Sparse version of summary.lm
A virtual Class: No objects may be created from it.
signature(x = "summary.mslm"): ...
Sparse version of summary.lm
A virtual Class: No objects may be created from it.
signature(x = "summary.slm"): ...
This is a design matrix arising from a bivariate smoothing problem using penalized triogram fitting. It is used in the SparseM vignette to illustrate the use of the sparse matrix image function.
data(triogramX)data(triogramX)
A 375 by 100 matrix stored in compressed sparse row format
Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R,
http://www.econ.uiuc.edu/~roger/research/home.html