"""
   lsqr v1.1.0 Ver.: OK (c) 01/10 - 12/19 Christos Iosifides 

   This program solves the least squares solution x that minimizes norm |A*x-B|

   Name:            lsqr
   Version:         1.1.0
   Copyright:       Ch Iossif @ 2010
   Author:          Christos Iosifidis
   Date:            20/01/10
   Modified:        05/02/10 11:38, 22/12/2019 13:15
   Description:     Solves the least squares solution x that minimizes norm |A*x-B|

   Input :
   
   M =  4 N =  2
   
   A = 
   [[ 0.588 -0.809]
    [-0.768 -0.64 ]
    [-0.404  0.915]
    [ 0.898  0.441]]
   
   B = 
   [-0.05  0.07  0.06 -0.08]
   
   Output :
   
   X = 
   [-0.09428331  0.00918387]
   
   sigma =  0.013424265794185002
   
   Vx = 
   [[ 0.00704931 -0.0001419 ]
    [-0.0001419   0.00640821]]
   

   Copyright (C) Dec 2009 Ch Iossif <chiossif@yahoo.com>
  
   This program is free software: you can redistribute it and/or modify
   it under the terms of the GNU General Public License as published by
   the Free Software Foundation, either version 3 of the License, or
   (at your option) any later version.
  
   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
  
   You should have received a copy of the GNU General Public License
   along with this program.  If not, see <http://www.gnu.org/licenses/>.
"""

import numpy as np
import math

M = int(input("M = "))
N = int(input("N = "))

A = []
for i in range(0, M):
	A_line = []
	for j in range(0, N):
		A_line.append(float(input())) # adding the element
	A.append(A_line) # adding the row into A matrix

B = []
for i in range(0, M):
	B.append(float(input())) # adding the element 

#M=4
#N=2
#A=[ [0.588, -0.809], [-0.768, -0.64], [-0.404, 0.915], [ 0.898, 0.441]]
#B=[ -0.05, 0.07, 0.06, -0.08]

# Python lists to numpy arrays
a=np.asarray(A,dtype=np.float64)
b=np.asarray(B,dtype=np.float64)
m=a.shape[0]
n=a.shape[1]

# Input
print("Input :")
print("\nM = ", m , "N = ", n)
print("\nA = ")
print(a)
print("\nB = ")
print(b)

# Do the math
at=np.transpose(a)
ata=np.matmul(at,a)
ata_1=np.linalg.inv(ata)
atb=np.matmul(at,b)
x=np.matmul(ata_1,atb)
y=b-np.matmul(a,x)
sigma=math.sqrt(np.sum(np.square(y))/(m-n))
vx=np.dot(ata_1,sigma)

# Output
print("\nOutput :")
print("\nX = ")
print(x)
print("\nsigma = ",sigma)
print("\nVx = ")
print(vx)