3.1

First method

Assume observation equation is:

where a_ii (i = 1,2,3)—constant, l_j (j = 1,2,3,4)—measurement result, which is also a constant. List the coefficient matrix A and the measurement result matrix L ^[3]

Eq.(1) shows that A^T A and A^T L should be known, then determine the inverse matrix C of A^T A:

The least square estimator of X̂ are obtained by multiplying C by AL.

3.2

Second method

First, make a table according to the observation equation, where a_ii (i = 1,2,3,4) and l_j (j = 1,2,3,4) are all four numbers of the corresponding columns of the constants Ai(i = 1,2,3) and Y, which are given by the four formulas above are assumed (Note: They are not a vector).

Next, make another form firstly. Ai × Aj in first row represents the four numbers, which multiply the corresponding the four numbers that are represented by Ai and Aj, respectively. For example, A1 × A1 means the number of a₁₁ × a₁₁、 a₂₁ × a₂₁、 a₃₁ × a₃₁、 a₄₁ × a₄₁. And ∑i(i = 1,2,3,…) in last row represent the sum of the first four digits of the corresponding column.

Table 1

Calculation table of normal equation coefficient

Then the last line of the table three per group divided into four groups and the three equations are written

Third, the least squares estimator could be calculated by solving the equations obtained in the second step.

Prove: The matrix X is grouped

So (a is a column vector of 4 × 1, b is a row vector of 1 × 4.), and so

According to the basic characteristics of transposed and matrix

If ∑1 = a1b1, the sum of the above table corresponds to the elements in the X^T X and X^T Y matrices. So the finally corresponding equation set in the third step is:

Forth, the standard deviation S of the measured value is also obtained first, and the value of d_jj is calculated by the equation set obtained by the above form

And the value of C₁ obtained by the above formula is the value of d₁₁. Similarly, replace the value of the right side of the equation by 0 1 0 when calculating the value of d₂₂, and replace the value of the right side of the equation by 0 0 1

when calculating the value of

Finally, the corresponding standard deviation is obtained by

Prove: In the calculation of the matrix of the inverse matrix, the defining equation is CC⁻¹ = E, from the above we already know

(This is a symmetric matrix, and its inverse matrix is symmetric matrix as well.) Assume its inverse matrix is , then E is a unit matrix of 3 × 3. By matrix division, whenC is multiplied by the first column of C⁻¹, the resulting value is the first column of E,

The values of t1、 t2、 t3 are the diagonal of the inverse matrix, that is, the value of d_jj which we need.