SGBCO - SBGCO factors a real band matrix by Gaussian elimination and
estimates the condition of the matrix.
If RCOND is not needed, SGBFA is slightly faster. To solve A*X = B ,
follow SBGCO by SGBSL. To compute INVERSE(A)*C , follow SBGCO by SGBSL.
To compute DETERMINANT(A) , follow SBGCO by SGBDI.
ABD REAL(LDA, N)
contains the matrix in band storage. The columns
of the matrix are stored in the columns of ABD and
the diagonals of the matrix are stored in rows
ML+1 through 2*ML+MU+1 of ABD .
See the comments below for details.
the leading dimension of the array ABD .
LDA must be .GE. 2*ML + MU + 1 .
the order of the original matrix.
number of diagonals below the main diagonal.
0 .LE. ML .LT. N .
number of diagonals above the main diagonal.
0 .LE. MU .LT. N .
More efficient if ML .LE. MU . On Return
ABD an upper triangular matrix in band storage and
the multipliers which were used to obtain it.
The factorization can be written A = L*U where
L is a product of permutation and unit lower
triangular matrices and U is upper triangular.
an integer vector of pivot indices.
an estimate of the reciprocal condition of A .
For the system A*X = B , relative perturbations
in A and B of size EPSILON may cause
relative perturbations in X of size EPSILON/RCOND .
If RCOND is so small that the logical expression
1.0 + RCOND .EQ. 1.0
is true, then A may be singular to working
precision. In particular, RCOND is zero if
exact singularity is detected or the estimate
a work vector whose contents are usually unimportant.
If A is close to a singular matrix, then Z is
an approximate null vector in the sense that
NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . Band Storage
If A is a band matrix, the following program segment
will set up the input.
ML = (band width below the diagonal)
MU = (band width above the diagonal)
M = ML + MU + 1
DO 20 J = 1, N
I1 = MAX0(1, J-MU)
I2 = MIN0(N, J+ML)
DO 10 I = I1, I2
K = I - J + M
ABD(K,J) = A(I,J)
This uses rows ML+1 through 2*ML+MU+1 of ABD .
In addition, the first ML rows in ABD are used for
elements generated during the triangularization.
The total number of rows needed in ABD is 2*ML+MU+1 .
The ML+MU by ML+MU upper left triangle and the
ML by ML lower right triangle are not referenced. Example: If the
original matrix is
11 12 13 0 0 0
21 22 23 24 0 0
0 32 33 34 35 0
0 0 43 44 45 46
0 0 0 54 55 56
0 0 0 0 65 66 then N = 6, ML = 1, MU = 2, LDA .GE. 5 and ABD
* * * + + + , * = not used
* * 13 24 35 46 , + = used for pivoting
* 12 23 34 45 56
11 22 33 44 55 66
21 32 43 54 65 * LINPACK. This version dated 08/14/78 . Cleve
Moler, University of New Mexico, Argonne National Lab. Subroutines and
Functions LINPACK SGBFA BLAS SAXPY,SDOT,SSCAL,SASUM Fortran
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