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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgbsvx (f07bp)

Purpose

nag_lapack_zgbsvx (f07bp) uses the LULU factorization to compute the solution to a complex system of linear equations
AX = B ,  ATX = B   or   AHX = B ,
AX=B ,  ATX=B   or   AHX=B ,
where AA is an nn by nn band matrix with klkl subdiagonals and kuku superdiagonals, and XX and BB are nn by rr matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[ab, afb, ipiv, equed, r, c, b, x, rcond, ferr, berr, rwork, info] = f07bp(fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, afb, ipiv, equed, r, c, b, x, rcond, ferr, berr, rwork, info] = nag_lapack_zgbsvx(fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgbsvx (f07bp) performs the following steps:
  1. Equilibration
    The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting fact = 'E'fact='E'. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems AX = B AX=B , ATX = B ATX=B  and AHX = B AHX=B  are
    (DRADC) (DC1X) = DR B ,
    ( DR A DC ) ( DC-1X ) = DR B ,
    (DRADC)T (DR1X) = DC B ,
    ( DR A DC )T ( DR-1 X ) = DC B ,
    and
    (DRADC)H (DR1X) = DC B ,
    ( DR A DC )H ( DR-1 X ) = DC B ,
    respectively, where DR DR  and DC DC  are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
    When equilibration is used, AA will be overwritten by DR A DC DR A DC  and BB will be overwritten by DR B DR B  (or DC B DC B  when the solution of ATX = B ATX=B  or AHX = B AHX=B  is sought).
  2. Factorization
    The matrix AA, or its scaled form, is copied and factored using the LULU decomposition
    A = PLU ,
    A=PLU ,
    where PP is a permutation matrix, LL is a unit lower triangular matrix, and UU is upper triangular.
    This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to nag_lapack_zgbsvx (f07bp) with the same matrix AA.
  3. Condition Number Estimation
    The LULU factorization of AA determines whether a solution to the linear system exists. If some diagonal element of UU is zero, then UU is exactly singular, no solution exists and the function returns with a failure. Otherwise the factorized form of AA is used to estimate the condition number of the matrix AA. If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.
  4. Solution
    The (equilibrated) system is solved for XX ( DC1X DC-1X  or DR1X DR-1X ) using the factored form of AA ( DRADC DRADC ).
  5. Iterative Refinement
    Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
  6. Construct Solution Matrix XX
    If equilibration was used, the matrix XX is premultiplied by DC DC  (if trans = 'N'trans='N') or DR DR  (if trans = 'T'trans='T' or 'C''C') so that it solves the original system before equilibration.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix AA is supplied on entry, and if not, whether the matrix AA should be equilibrated before it is factorized.
fact = 'F'fact='F'
afb and ipiv contain the factorized form of AA. If equed'N'equed'N', the matrix AA has been equilibrated with scaling factors given by r and c. ab, afb and ipiv are not modified.
fact = 'N'fact='N'
The matrix AA will be copied to afb and factorized.
fact = 'E'fact='E'
The matrix AA will be equilibrated if necessary, then copied to afb and factorized.
Constraint: fact = 'F'fact='F', 'N''N' or 'E''E'.
2:     trans – string (length ≥ 1)
Specifies the form of the system of equations.
trans = 'N'trans='N'
AX = BAX=B (No transpose).
trans = 'T'trans='T'
ATX = BATX=B (Transpose).
trans = 'C'trans='C'
AHX = BAHX=B (Conjugate transpose).
Constraint: trans = 'N'trans='N', 'T''T' or 'C''C'.
3:     kl – int64int32nag_int scalar
klkl, the number of subdiagonals within the band of the matrix AA.
Constraint: kl0kl0.
4:     ku – int64int32nag_int scalar
kuku, the number of superdiagonals within the band of the matrix AA.
Constraint: ku0ku0.
5:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least kl + ku + 1kl+ku+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn coefficient matrix AA.
The matrix is stored in rows 11 to kl + ku + 1kl+ku+1, more precisely, the element AijAij must be stored in
ab(ku + 1 + ij,j)  for ​max (1,jku)imin (n,j + kl).
abku+1+i-jj  for ​max(1,j-ku)imin(n,j+kl).
See Section [Further Comments] for further details.
If fact = 'F'fact='F' and equed'N'equed'N', AA must have been equilibrated by the scaling factors in r and/or c.
6:     afb(ldafb, : :) – complex array
The first dimension of the array afb must be at least 2 × kl + ku + 12×kl+ku+1
The second dimension of the array must be at least max (1,n)max(1,n)
If fact = 'N'fact='N' or 'E''E', afb need not be set.
If fact = 'F'fact='F', details of the LULU factorization of the nn by nn band matrix AA, as computed by nag_lapack_zgbtrf (f07br).
The upper triangular band matrix UU, with kl + kukl+ku superdiagonals, is stored in rows 11 to kl + ku + 1kl+ku+1 of the array, and the multipliers used to form the matrix LL are stored in rows kl + ku + 2kl+ku+2 to 2kl + ku + 12kl+ku+1.
If equed'N'equed'N', afb is the factorized form of the equilibrated matrix AA.
7:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
If fact = 'N'fact='N' or 'E''E', ipiv need not be set.
If fact = 'F'fact='F', ipiv contains the pivot indices from the factorization A = LUA=LU, as computed by nag_lapack_dgbtrf (f07bd); row ii of the matrix was interchanged with row ipiv(i)ipivi.
8:     equed – string (length ≥ 1)
If fact = 'N'fact='N' or 'E''E', equed need not be set.
If fact = 'F'fact='F', equed must specify the form of the equilibration that was performed as follows:
  • if equed = 'N'equed='N', no equilibration;
  • if equed = 'R'equed='R', row equilibration, i.e., AA has been premultiplied by DRDR;
  • if equed = 'C'equed='C', column equilibration, i.e., AA has been postmultiplied by DCDC;
  • if equed = 'B'equed='B', both row and column equilibration, i.e., AA has been replaced by DRADCDRADC.
Constraint: if fact = 'F'fact='F', equed = 'N'equed='N', 'R''R', 'C''C' or 'B''B'.
9:     r( : :) – double array
Note: the dimension of the array r must be at least max (1,n)max(1,n).
If fact = 'N'fact='N' or 'E''E', r need not be set.
If fact = 'F'fact='F' and equed = 'R'equed='R' or 'B''B', r must contain the row scale factors for AA, DRDR; each element of r must be positive.
10:   c( : :) – double array
Note: the dimension of the array c must be at least max (1,n)max(1,n).
If fact = 'N'fact='N' or 'E''E', c need not be set.
If fact = 'F'fact='F' or equed = 'C'equed='C' or 'B''B', c must contain the column scale factors for AA, DCDC; each element of c must be positive.
11:   b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr right-hand side matrix BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The second dimension of the arrays ab, afb, ipiv, r, c.
nn, the number of linear equations, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
rr, the number of right-hand sides, i.e., the number of columns of the matrix BB.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldab ldafb ldb ldx work

Output Parameters

1:     ab(ldab, : :) – complex array
The first dimension of the array ab will be kl + ku + 1kl+ku+1
The second dimension of the array will be max (1,n)max(1,n)
ldabkl + ku + 1ldabkl+ku+1.
If fact = 'F'fact='F' or 'N''N', or if fact = 'E'fact='E' and equed = 'N'equed='N', ab is not modified.
If equed'N'equed'N' then, if no constraints are violated, AA is scaled as follows:
  • if equed = 'R'equed='R', A = DrAA=DrA;
  • if equed = 'C'equed='C', A = ADcA=ADc;
  • if equed = 'B'equed='B', A = DrADcA=DrADc.
2:     afb(ldafb, : :) – complex array
The first dimension of the array afb will be 2 × kl + ku + 12×kl+ku+1
The second dimension of the array will be max (1,n)max(1,n)
ldafb2 × kl + ku + 1ldafb2×kl+ku+1.
If fact = 'F'fact='F', afb is unchanged from entry.
Otherwise, if no constraints are violated, then if fact = 'N'fact='N', afb returns details of the LULU factorization of the band matrix AA, and if fact = 'E'fact='E', afb returns details of the LULU factorization of the equilibrated band matrix AA (see the description of ab for the form of the equilibrated matrix).
3:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
If fact = 'F'fact='F', ipiv is unchanged from entry.
Otherwise, if no constraints are violated, ipiv contains the pivot indices that define the permutation matrix PP; at the iith step row ii of the matrix was interchanged with row ipiv(i)ipivi. ipiv(i) = iipivi=i indicates a row interchange was not required.
If fact = 'N'fact='N', the pivot indices are those corresponding to the factorization A = LUA=LU of the original matrix AA.
If fact = 'E'fact='E', the pivot indices are those corresponding to the factorization of A = LUA=LU of the equilibrated matrix AA.
4:     equed – string (length ≥ 1)
If fact = 'F'fact='F', equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of equilibration that was performed as specified above.
5:     r( : :) – double array
Note: the dimension of the array r must be at least max (1,n)max(1,n).
If fact = 'F'fact='F', r is unchanged from entry.
Otherwise, if no constraints are violated and equed = 'R'equed='R' or 'B''B', r contains the row scale factors for AA, DRDR, such that AA is multiplied on the left by DRDR; each element of r is positive.
6:     c( : :) – double array
Note: the dimension of the array c must be at least max (1,n)max(1,n).
If fact = 'F'fact='F', c is unchanged from entry.
Otherwise, if no constraints are violated and equed = 'C'equed='C' or 'B''B', c contains the row scale factors for AA, DCDC; each element of c is positive.
7:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
If equed = 'N'equed='N', b is not modified.
If trans = 'N'trans='N' and equed = 'R'equed='R' or 'B''B', b stores DRBDRB.
If trans = 'T'trans='T' or 'C''C' and equed = 'C'equed='C' or 'B''B', b stores DCBDCB.
8:     x(ldx, : :) – complex array
The first dimension of the array x will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldxmax (1,n)ldxmax(1,n).
If INFO = 0INFO=0 or n + 1n+1, the nn by rr solution matrix XX to the original system of equations. Note that the arrays AA and BB are modified on exit if equed'N'equed'N', and the solution to the equilibrated system is DC1XDC-1X if trans = 'N'trans='N' and equed = 'C'equed='C' or 'B''B', or DR1XDR-1X if trans = 'T'trans='T' or 'C''C' and equed = 'R'equed='R' or 'B''B'.
9:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal condition number of the matrix AA (after equilibration if that is performed), computed as rcond = 1.0 / (A1A11)rcond=1.0/(A1 A-11 ).
10:   ferr(nrhs_p) – double array
If INFO = 0INFO=0 or n + 1n+1, an estimate of the forward error bound for each computed solution vector, such that jxj / xjferr(j)x^j-xj/xjferrj where jx^j is the jjth column of the computed solution returned in the array x and xjxj is the corresponding column of the exact solution XX. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
11:   berr(nrhs_p) – double array
If INFO = 0INFO=0 or n + 1n+1, an estimate of the component-wise relative backward error of each computed solution vector jx^j (i.e., the smallest relative change in any element of AA or BB that makes jx^j an exact solution).
12:   rwork(max (1,n)max(1,n)) – double array
If INFO = 0INFO=0, rwork(1)rwork1 contains the reciprocal pivot growth factor max|aij| / max|uij|max|aij|/max|uij|. If rwork(1)rwork1 is much less than 11, then the stability of the LULU factorization of the (equilibrated) matrix AA could be poor. This also means that the solution XX, condition estimator rcond, and forward error bound ferr could be unreliable. If the factorization fails with INFO > 0andINFOnINFO>0andINFOn, rwork(1)rwork1 contains the reciprocal pivot growth factor for the leading info columns of AA.
13:   info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: fact, 2: trans, 3: n, 4: kl, 5: ku, 6: nrhs_p, 7: ab, 8: ldab, 9: afb, 10: ldafb, 11: ipiv, 12: equed, 13: r, 14: c, 15: b, 16: ldb, 17: x, 18: ldx, 19: rcond, 20: ferr, 21: berr, 22: work, 23: rwork, 24: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0andINFONINFO>0andINFON
If info = iinfo=i, uiiuii is exactly zero. The factorization has been completed, but the factor UU is exactly singular, so the solution and error bounds could not be computed. rcond = 0.0rcond=0.0 is returned.
W INFO = N + 1INFO=N+1
The triangular matrix UU is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector bb, the computed solution x^ is the exact solution of a perturbed system of equations (A + E) = b(A+E)x^=b, where
|E|c(n)εP|L||U| ,
|E|c(n)εP|L||U| ,
c(n)c(n) is a modest linear function of nn, and εε is the machine precision. See Section 9.3 of Higham (2002) for further details.
If xx is the true solution, then the computed solution x^ satisfies a forward error bound of the form
( x )/( ) wc cond(A,,b)
x-x^ x^ wc cond(A,x^,b)
where cond(A,,b) = |A1|(|A||| + |b|) / cond(A) = |A1||A|κ (A) cond(A,x^,b) = |A-1| ( |A| |x^| + |b| ) / x^ cond(A) = |A-1| |A| κ (A). If x^  is the j j th column of X X , then wc wc  is returned in berr(j) berrj  and a bound on x / x - x^ / x^  is returned in ferr(j) ferrj . See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n = 6 n=6 , kl = 1 kl=1 , and ku = 2 ku=2 . Storage of the band matrix A A  in the array ab:
* * a13 a24 a35 a46
* a12 a23 a34 a45 a56
a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *
The total number of floating point operations required to solve the equations AX = B AX=B  depends upon the pivoting required, but if nkl + ku nkl + ku  then it is approximately bounded by O( n kl ( kl + ku ) ) O( n kl ( kl + ku ) )  for the factorization and O( n (2kl + ku) r ) O( n ( 2 kl + ku ) r )  for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see nag_lapack_zgbrfs (f07bv) for information on the floating point operations required.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of this function is nag_lapack_dgbsvx (f07bb).

Example

function nag_lapack_zgbsvx_example
fact = 'Equilibration';
trans = 'No transpose';
kl = int64(1);
ku = int64(2);
ab = [0,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
      0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
      0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
afb = complex(zeros(5, 4) );
ipiv = [int64(0);0;0;0];
equed = ' ';
r = [0;
     0;
     0;
     0];
c = [0;
     0;
     0;
     0];
b = [ -1.06 + 21.5i,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[abOut, afbOut, ipivOut, equedOut, rOut, cOut, bOut, x, ...
 rcond, ferr, berr, rwork, info] = ...
    nag_lapack_zgbsvx(fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.9700 - 2.8400i   0.5900 - 0.4800i
   0.0000 + 0.0000i  -2.0500 - 0.8500i  -3.9900 + 4.0100i   3.3300 - 1.0400i
  -1.6500 + 2.2600i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -0.4600 - 1.7200i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.4800 - 1.0900i   0.0000 + 0.0000i


afbOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.5900 - 0.4800i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -3.9900 + 4.0100i   3.3300 - 1.0400i
   0.0000 + 0.0000i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -1.7692 - 1.8587i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.9303 - 3.0086i   0.4338 + 0.1233i
   0.3587 + 0.2619i   0.2314 + 0.6358i   0.7604 + 0.2429i   0.0000 + 0.0000i


ipivOut =

                    2
                    3
                    3
                    4


equedOut =

N


rOut =

    0.2558
    0.1250
    0.2288
    0.1795


cOut =

    1.0000
    1.2139
    1.0000
    1.0000


bOut =

  -1.0600 +21.5000i  12.8500 + 2.8400i
 -22.7200 -53.9000i -70.2200 +21.5700i
  28.2400 -38.6000i -20.7300 - 1.2300i
 -34.5600 +16.7300i  26.0100 +31.9700i


x =

  -3.0000 + 2.0000i   1.0000 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


rcond =

    0.0096


ferr =

   1.0e-13 *

    0.3607
    0.4350


berr =

   1.0e-15 *

    0.0603
    0.1011


rwork =

    1.0000
    0.0000
    0.0000
    0.0000


info =

                    0


function f07bp_example
fact = 'Equilibration';
trans = 'No transpose';
kl = int64(1);
ku = int64(2);
ab = [0,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
      0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
      0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
afb = complex(zeros(5, 4) );
ipiv = [int64(0);0;0;0];
equed = ' ';
r = [0;
     0;
     0;
     0];
c = [0;
     0;
     0;
     0];
b = [ -1.06 + 21.5i,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[abOut, afbOut, ipivOut, equedOut, rOut, cOut, bOut, x, ...
 rcond, ferr, berr, rwork, info] = ...
    f07bp(fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.9700 - 2.8400i   0.5900 - 0.4800i
   0.0000 + 0.0000i  -2.0500 - 0.8500i  -3.9900 + 4.0100i   3.3300 - 1.0400i
  -1.6500 + 2.2600i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -0.4600 - 1.7200i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.4800 - 1.0900i   0.0000 + 0.0000i


afbOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.5900 - 0.4800i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -3.9900 + 4.0100i   3.3300 - 1.0400i
   0.0000 + 0.0000i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -1.7692 - 1.8587i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.9303 - 3.0086i   0.4338 + 0.1233i
   0.3587 + 0.2619i   0.2314 + 0.6358i   0.7604 + 0.2429i   0.0000 + 0.0000i


ipivOut =

                    2
                    3
                    3
                    4


equedOut =

N


rOut =

    0.2558
    0.1250
    0.2288
    0.1795


cOut =

    1.0000
    1.2139
    1.0000
    1.0000


bOut =

  -1.0600 +21.5000i  12.8500 + 2.8400i
 -22.7200 -53.9000i -70.2200 +21.5700i
  28.2400 -38.6000i -20.7300 - 1.2300i
 -34.5600 +16.7300i  26.0100 +31.9700i


x =

  -3.0000 + 2.0000i   1.0000 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


rcond =

    0.0096


ferr =

   1.0e-13 *

    0.3607
    0.4350


berr =

   1.0e-15 *

    0.0603
    0.1011


rwork =

    1.0000
    0.0000
    0.0000
    0.0000


info =

                    0



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