用R解欠定线性系统 [英] Solving underdetermined linear systems with R
问题描述
R 可以解决欠定线性系统:
R can solve underdetermined linear systems:
A = matrix((1:12)^2,3,4,T)
B = 1:3
qr(A)$rank # 3
qr.solve(A, B) # solutions will have one zero, not necessarily the same one
# 0.1875 -0.5000 0.3125 0.0000
solve(qr(A, LAPACK = TRUE), B)
# 0.08333333 -0.18750000 0.00000000 0.10416667
(它给出了无穷多个解).
(It gives one solution among the infinity of solutions).
但是,如果排名(此处为2)低于行数(此处为3),则它将行不通:
However, if the rank (here 2) is lower than the number of rows (here 3), it won't work:
A = matrix(c((1:8)^2,0,0,0,0),3,4,T)
B = c(1,2,0)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 4 9 16
# [2,] 25 36 49 64
# [3,] 0 0 0 0
qr.solve(A, B) # Error in qr.solve(A, B) : singular matrix
solve(qr(A, LAPACK = TRUE), B) # Error in qr.coef(a, b) : error code 3
但是该系统确实有解决方案!
but this system does have a solution!
I know that the general solution is to use SVD or generalized/pseudo inverse of A (see this question and its answers), but:
是否存在 solve
或 qr.solve
的平均值,以自动将系统AX = B减少为仅等级(A的等效系统CX = D)行, qr.solve(C,D)
可以直接使用吗?
Is there a mean with solve
or qr.solve
to automatically reduce the system AX=B to an equivalent system CX=D of only rank(A) rows, for which qr.solve(C, D)
would simply work out-of-the-box?
示例:
C = matrix(c((1:8)^2),2,4,T)
D = c(1,2)
qr.solve(C, D)
# -0.437500 0.359375 0.000000 0.000000
推荐答案
qr.coef
和 qr
似乎可以完成这项工作:
qr.coef
along with qr
seem to do the job:
(A <- matrix(c((1:8)^2, 0, 0, 0, 0), nrow = 3, ncol = 4, byrow = TRUE))
# [,1] [,2] [,3] [,4]
# [1,] 1 4 9 16
# [2,] 25 36 49 64
# [3,] 0 0 0 0
(B <- c(1, 2, 0))
# [1] 1 2 0
(X0 <- qr.coef(qr(A), B))
# [1] -0.437500 0.359375 NA NA
X0[is.na(X0)] <- 0
X0
# [1] -0.437500 0.359375 0.000000 0.000000
# Verification:
A %*% X0
# [,1]
# [1,] 1
# [2,] 2
# [3,] 0
第二个例子:
Second example:
(A<-matrix(c(1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0), nrow = 3, ncol = 4, byrow = TRUE))
# [,1] [,2] [,3] [,4]
# [1,] 1 2 0 0
# [2,] 1 2 0 0
# [3,] 1 2 1 0
(B<-c(1, 1, 2))
# [1] 1 1 2
qr.solve(A, B)
# Error in qr.solve(A, B) : singular matrix 'a' in solve
(X0 <- qr.coef(qr(A), B))
# [1] 1 NA 1 NA
X0[is.na(X0)] <- 0
X0
# [1] 1 0 1 0
A %*% X0
# [,1]
# [1,] 1
# [2,] 1
# [3,] 2
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