Question

R可以求解不确定线性系统：

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）。

但是，如果排名（此处为2）低于行数（此处为3），则它将行不通：

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！

我知道一般的解决方案是使用SVD或A的广义/伪逆（请参见this question及其答案），但是：

是否存在solve或qr.solve的平均值，以将系统AX = B自动降低为仅排名（A）行的等效系统CX = D 哪个qr.solve(C, D)可以直接使用？

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

Answer 1

qr.coef和qr似乎可以胜任：

(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

第二个示例：

(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

用R解欠定线性系统

1 个答案: