Matrix Norms

Computes a matrix norm of x, using Lapack for dense matrices. The norm can be the one ("O", or "1") norm, the infinity ("I") norm, the Frobenius ("F") norm, the maximum modulus ("M") among elements of a matrix, or the spectral norm or 2-norm ("2"), as determined by the value of type.

norm(x, type, ...)

Arguments

x

a real or complex matrix.

type

A character indicating the type of norm desired.

"O", "o" or "1"

specifies the one norm, (maximum absolute column sum);

"I" or "i"

specifies the infinity norm (maximum absolute row sum);

"F" or "f"

specifies the Frobenius norm (the Euclidean norm of x treated as if it were a vector);

"M" or "m"

specifies the maximum modulus of all the elements in x; and

"2"

specifies the “spectral norm” aka “2-norm”, which is the largest singular value (svd) of x.

The default is "O". Only the first character of type[1] is used.

...

further arguments passed to or from other methods.

Value

A numeric value of class "norm", representing the quantity chosen according to type.

Details

For dense matrices, the methods eventually call the Lapack functions dlange, dlansy, dlantr, zlange, zlansy, and zlantr.

See also

onenormest(), an approximate randomized estimate of the 1-norm condition number, efficient for large sparse matrices.

The norm() function from R's base package.

References

Anderson, E., et al. (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.

Examples

x <- Hilbert(9)
norm(x)# = "O" = "1"
#> [1] 2.828968
stopifnot(identical(norm(x), norm(x, "1")))
norm(x, "I")# the same, because 'x' is symmetric
#> [1] 2.828968

allnorms <- function(x) {
    ## norm(NA, "2") did not work until R 4.0.0
    do2 <- getRversion() >= "4.0.0" || !anyNA(x)
    vapply(c("1", "I", "F", "M", if(do2) "2"), norm, 0, x = x)
}
allnorms(x)
#>        1        I        F        M        2 
#> 2.828968 2.828968 1.755872 1.000000 1.725883 
allnorms(Hilbert(10))
#>        1        I        F        M        2 
#> 2.928968 2.928968 1.785527 1.000000 1.751920 

i <- c(1,3:8); j <- c(2,9,6:10); x <- 7 * (1:7)
A <- sparseMatrix(i, j, x = x)                      ##  8 x 10 "dgCMatrix"
(sA <- sparseMatrix(i, j, x = x, symmetric = TRUE)) ## 10 x 10 "dsCMatrix"
#> 10 x 10 sparse Matrix of class "dsCMatrix"
#>                                  
#>  [1,] . 7  .  .  .  .  .  .  .  .
#>  [2,] 7 .  .  .  .  .  .  .  .  .
#>  [3,] . .  .  .  .  .  .  . 14  .
#>  [4,] . .  .  .  . 21  .  .  .  .
#>  [5,] . .  .  .  .  . 28  .  .  .
#>  [6,] . .  . 21  .  .  . 35  .  .
#>  [7,] . .  .  . 28  .  .  . 42  .
#>  [8,] . .  .  .  . 35  .  .  . 49
#>  [9,] . . 14  .  .  . 42  .  .  .
#> [10,] . .  .  .  .  .  . 49  .  .
(tA <- sparseMatrix(i, j, x = x, triangular= TRUE)) ## 10 x 10 "dtCMatrix"
#> 10 x 10 sparse Matrix of class "dtCMatrix"
#>                               
#>  [1,] . 7 . . .  .  .  .  .  .
#>  [2,] . . . . .  .  .  .  .  .
#>  [3,] . . . . .  .  .  . 14  .
#>  [4,] . . . . . 21  .  .  .  .
#>  [5,] . . . . .  . 28  .  .  .
#>  [6,] . . . . .  .  . 35  .  .
#>  [7,] . . . . .  .  .  . 42  .
#>  [8,] . . . . .  .  .  .  . 49
#>  [9,] . . . . .  .  .  .  .  .
#> [10,] . . . . .  .  .  .  .  .
(allnorms(A) -> nA)
#> Warning: 'norm' via sparse -> dense coercion
#>        1        I        F        M        2 
#> 56.00000 49.00000 82.82512 49.00000 49.00000 
allnorms(sA)
#> Warning: 'norm' via sparse -> dense coercion
#>         1         I         F         M         2 
#>  84.00000  84.00000 117.13240  49.00000  61.54212 
allnorms(tA)
#> Warning: 'norm' via sparse -> dense coercion
#>        1        I        F        M        2 
#> 56.00000 49.00000 82.82512 49.00000 49.00000 
stopifnot(all.equal(nA, allnorms(as(A, "matrix"))),
    all.equal(nA, allnorms(tA))) # because tA == rbind(A, 0, 0)
#> Warning: 'norm' via sparse -> dense coercion
A. <- A; A.[1,3] <- NA
stopifnot(is.na(allnorms(A.))) # gave error
#> Warning: 'norm' via sparse -> dense coercion