DGBCO(3F) DGBCO(3F)
DGBCO - DGBCO factors a double precision band matrix by Gaussian
elimination and estimates the condition of the matrix.
If RCOND is not needed, DGBFA is slightly faster. To solve A*X = B ,
follow DGBCO by DGBSL. To compute INVERSE(A)*C , follow DGBCO by DGBSL.
To compute DETERMINANT(A) , follow DGBCO by DGBDI.
SUBROUTINE DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)
On Entry
ABD DOUBLE PRECISION(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.
LDA INTEGER
the leading dimension of the array ABD .
LDA must be .GE. 2*ML + MU + 1 .
N INTEGER
the order of the original matrix.
ML INTEGER
number of diagonals below the main diagonal.
0 .LE. ML .LT. N .
MU INTEGER
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.
IPVT INTEGER(N)
an integer vector of pivot indices.
RCOND DOUBLE PRECISION
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 .
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DGBCO(3F) DGBCO(3F)
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
underflows.
Z DOUBLE PRECISION(N)
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)
10 CONTINUE
20 CONTINUE
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
should contain
* * * + + + , * = 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 used: LINPACK DGBFA BLAS DAXPY,DDOT,DSCAL,DASUM Fortran
DABS,DMAX1,MAX0,MIN0,DSIGN
PPPPaaaaggggeeee 2222 [ Back ]
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