clatbs(3)

NAME

CLATBS - solve one of the triangular systems A * x = s*b,

A**T * x = s*b, or A**H * x = s*b,

SYNOPSIS

SUBROUTINE  CLATBS(  UPLO, TRANS, DIAG, NORMIN, N, KD, AB,
LDAB, X, SCALE, CNORM, INFO )
    CHARACTER      DIAG, NORMIN, TRANS, UPLO
    INTEGER        INFO, KD, LDAB, N
    REAL           SCALE
    REAL           CNORM( * )
    COMPLEX        AB( LDAB, * ), X( * )

PURPOSE

CLATBS solves one of the triangular systems A * x = s*b,

A**T * x = s*b, or A**H * x = s*b, with scaling to prevent over

flow, where A is an upper or lower triangular band matrix. Here

A' denotes the transpose of A, x and b are n-element 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 thresh

old. If the unscaled problem will not cause overflow, the Level

2 BLAS routine CTBSV is called. If the matrix A is singular

(A(j,j) = 0 for some j), then s is set to 0 and a non-trivial so

lution 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 trans

pose)

DIAG (input) CHARACTER*1

Specifies whether or not the matrix A is unit tri

angular. = 'N': Non-unit 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 array, dimension (LDAB,N)

The upper or lower triangular band matrix A,

stored in the first KD+1 rows of the array. The j-th column of A

is stored in the j-th column of the array AB as follows: if UPLO

= 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO =

'L', AB(1+i-j,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 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) REAL

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 ex

act or approximate solution to A*x = 0.

CNORM (input or output) REAL array, dimension (N)

If NORMIN = 'Y', CNORM is an input argument and

CNORM(j) contains the norm of the off-diagonal part of the j-th

column of A. If TRANS = 'N', CNORM(j) must be greater than or

equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)

must be greater than or equal to the 1-norm.

If NORMIN = 'N', CNORM is an output argument and

CNORM(j) returns the 1-norm of the offdiagonal part of the j-th

column of A.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -k, the k-th argument had an ille

gal value

FURTHER DETAILS

A rough bound on x is computed; if that is less than over

flow, CTBSV is called, otherwise, specific code is used which

checks for possible overflow or divide-by-zero at every opera

tion.

A columnwise scheme is used for solving A*x = b. The ba

sic 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 infinity

norm 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 CTB

SV 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 non-trivial solution to

A*x = 0 is found.

Similarly, a row-wise 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:j-1,j]' * x[1:j-1] ) /

A(j,j)

end

We simultaneously compute two bounds

G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ),

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(j-1) and M(j) >=

M(j-1) for j >= 1. Then the bound on x(j) is

M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j)

<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)

)

1<=i<=j

and we can safely call CTBSV if 1/M(n) and 1/G(n) are both

greater than max(underflow, 1/overflow).

LAPACK version 3.0 15 June 2000

CLATBS(3)