SYNOPSIS
 SUBROUTINE ZLATBS(
 UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO )
 CHARACTER DIAG, NORMIN, TRANS, UPLO
 INTEGER INFO, KD, LDAB, N
 DOUBLE PRECISION SCALE
 DOUBLE PRECISION CNORM( * )
 COMPLEX*16 AB( LDAB, * ), X( * )
PURPOSE
ZLATBS solves one of the triangular systems with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A' denotes the transpose of A, x and b are nelement vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a nontrivial solution to A*x = 0 is returned.ARGUMENTS
 UPLO (input) CHARACTER*1

Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular  TRANS (input) CHARACTER*1

Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T * x = s*b (Transpose)
= 'C': Solve A**H * x = s*b (Conjugate transpose)  DIAG (input) CHARACTER*1

Specifies whether or not the matrix A is unit triangular.
= 'N': Nonunit triangular
= 'U': Unit triangular  NORMIN (input) CHARACTER*1

Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.  N (input) INTEGER
 The order of the matrix A. N >= 0.
 KD (input) INTEGER
 The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.
 AB (input) COMPLEX*16 array, dimension (LDAB,N)
 The upper or lower triangular band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd).
 LDAB (input) INTEGER
 The leading dimension of the array AB. LDAB >= KD+1.
 X (input/output) COMPLEX*16 array, dimension (N)
 On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.
 SCALE (output) DOUBLE PRECISION
 The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.
 CNORM (input or output) DOUBLE PRECISION array, dimension (N)
 If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the offdiagonal part of the jth column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinitynorm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1norm of the offdiagonal part of the jth column of A.
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = k, the kth argument had an illegal value
FURTHER DETAILS
A rough bound on x is computed; if that is less than overflow, ZTBSV is called, otherwise, specific code is used which checks for possible overflow or dividebyzero at every operation.A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n]  x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) /  A(j+1,j+1) 
G(j+1) <= G(j) + M(j+1) *  A[j+2:n,j+1] 
<= G(j) ( 1 + CNORM(j+1) /  A(j+1,j+1)  )
where CNORM(j+1) is greater than or equal to the infinitynorm of column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) /  A(i,i)  )
1<=i<=j
and
x(j) <= ( G(0) / A(j,j) ) product ( 1 + CNORM(i) / A(i,i) )
1<=i< j
Since x(j) <= M(j), we use the Level 2 BLAS routine ZTBSV if the reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a nontrivial solution to A*x = 0 is found. Similarly, a rowwise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j)  A[1:j1,j]' * x[1:j1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i)  A[1:i1,i]' * x[1:i1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j1) * ( 1 + CNORM(j) ) /  A(j,j) 
<= M(0) * product ( ( 1 + CNORM(i) ) / A(i,i) )
1<=i<=j
and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).