This formula implies omin(B)<Ip1illp221...Pnnl, i.e., (A)S△(A) 0 Remark.Since the singular values and Pnn are continuous functions of the matrix entries,hence if pnn-0 then amin(B)-0.The converse is also true.It also can be seen that Ae(A)and A(A)change continuously with e>0. 4 Numerical experiments Now let us calculate pseudospectra properly.The place to begin is with the column pivoting QR decomposition.In numerical experiments,we observe that,if pk>>1,k=1,2,...,n then G,b21.In order to avoid this situation,we modify our formula G2 e as Gb1,62,..,onl/pil<e/p1 Hence,the algorithm is to compute column pivoting QR de- composition of zI-A for values of z on a grid in the plane and then generate a contour plot from this data.At last,we also notice that if Gb,b2,...,on]e then we get omin(B)e, but the converse may not be true (see Lawson and Hanson[5]p31). Algorithm5.1 (1)For each ze grid computing B=2I-A=[b1;...o]; (2)Computing the column pivoting QR decomposition of B; (3)IfG[b1,b2,,bnl/lp1l≤e/IPul then z∈c(A) else goto step (1). Now we present some numerical examples to examine our conclusions. Examplel We denote the matrix A=rand(5,5), 0.19340.69790.49660.6602 0.7271 0.68220.37840.8998 0.34200.3093 A= 0.30280.86000.82160.28970.8385 0.54170.85370.64490.34120.5681 0.15090.59360.81800.53410.3704 Figure 1 shows the pseudospectra of A,which the eigenvalue drawn as dots.Note that the sets are nested as indicated in Theorem3.1.In figure2,we see the 5 x 4 matrix of A(:,1:4).The inclusion properties of Theorem3.1(iii)can be clearly seen that the pseudospectra of the square matrix A are bigger than those of A(:,1:4). 6This formula implies σ n min(B) ≤ |ρ11||ρ22|...|ρnn|, i.e., Λ²(A) ⊆ Λ ² 1 n (A). ut Remark. Since the singular values and ρnn are continuous functions of the matrix entries, hence if ρnn −→ 0 then σmin(B) −→ 0. The converse is also true. It also can be seen that Λ²(A) and Λ²(A) change continuously with ² > 0. 4 Numerical experiments Now let us calculate pseudospectra properly. The place to begin is with the column pivoting QR decomposition. In numerical experiments,we observe that, if |ρkk| >> 1, k = 1, 2, ..., n then G 1 2 [b1, b2, ..., bn] >> 1. In order to avoid this situation, we modify our formula G 1 2 [b1, b2, ..., bn] ≤ ² as G 1 2 [b1, b2, ..., bn]/|ρ11| ≤ ²/|ρ11| Hence, the algorithm is to compute column pivoting QR de￾composition of z ˜I − A for values of z on a grid in the plane and then generate a contour plot from this data. At last, we also notice that if G 1 2 [b1, b2, ..., bn] ≤ ² n then we get σmin(B) ≤ ², but the converse may not be true (see Lawson and Hanson[5]p31). Algorithm5.1 (1)For each z ∈ grid computing B = z ˜I − A = [b1, ..., bn]; (2) Computing the column pivoting QR decomposition of B; (3) If G 1 2 [b1, b2, ..., bn]/|ρ11| ≤ ²/|ρ11| then z ∈ Λ²(A) else goto step (1). Now we present some numerical examples to examine our conclusions. Example1 We denote the matrix A=rand(5,5), A =   0.1934 0.6979 0.4966 0.6602 0.7271 0.6822 0.3784 0.8998 0.3420 0.3093 0.3028 0.8600 0.8216 0.2897 0.8385 0.5417 0.8537 0.6449 0.3412 0.5681 0.1509 0.5936 0.8180 0.5341 0.3704   . Figure 1 shows the pseudospectra of A,which the eigenvalue drawn as dots. Note that the sets are nested as indicated in Theorem3.1.In figure2, we see the 5 × 4 matrix of A(:,1:4). The inclusion properties of Theorem3.1(iii) can be clearly seen that the pseudospectra of the square matrix A are bigger than those of A(:,1:4). 5