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### Contents

DLATBS(3F)							    DLATBS(3F)

### NAME[Toc][Back]

DLATBS - solve one	of the triangular systems   A *x = s*b or A'*x = s*b
with scaling to prevent overflow, where A is an upper or lower triangular
band matrix

### SYNOPSIS[Toc][Back]

SUBROUTINE	DLATBS(	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 AB( LDAB, * ), CNORM(	* ), X(	* )

### PURPOSE[Toc][Back]

DLATBS solves one of the triangular systems 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 threshold.	If the
unscaled problem will not cause overflow, the Level 2 BLAS	routine	DTBSV
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 solution to A*x = 0 is returned.

### ARGUMENTS[Toc][Back]

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'* x = s*b	(Transpose)
= 'C':  Solve A'* x = s*b	(Conjugate transpose = Transpose)

DIAG    (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular.	= '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.

Page 1

DLATBS(3F)							    DLATBS(3F)

KD	     (input) INTEGER
The number	of subdiagonals	or superdiagonals in the triangular
matrix A.	KD >= 0.

AB	     (input) DOUBLE PRECISION 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 jth
column of the array AB as follows:  if UPLO = 'U', AB(kd+1+ij,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) DOUBLE PRECISION 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	 or
A'* 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 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 illegal value

FURTHER	DETAILS
A rough bound on x	is computed; if	that is	less than overflow, DTBSV is
called, otherwise,	specific code is used which checks for possible
overflow or divide-by-zero	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

Page 2

DLATBS(3F)							    DLATBS(3F)

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 DTBSV 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'*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 DTBSV if 1/M(n) and	1/G(n) are both	greater	than
max(underflow, 1/overflow).

Page 3

DLATBS(3F)							    DLATBS(3F)
DLATBS(3F)							    DLATBS(3F)

### NAME[Toc][Back]

DLATBS - solve one	of the triangular systems   A *x = s*b or A'*x = s*b
with scaling to prevent overflow, where A is an upper or lower triangular
band matrix

### SYNOPSIS[Toc][Back]

SUBROUTINE	DLATBS(	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 AB( LDAB, * ), CNORM(	* ), X(	* )

### PURPOSE[Toc][Back]

DLATBS solves one of the triangular systems 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 threshold.	If the
unscaled problem will not cause overflow, the Level 2 BLAS	routine	DTBSV
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 solution to A*x = 0 is returned.

### ARGUMENTS[Toc][Back]

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'* x = s*b	(Transpose)
= 'C':  Solve A'* x = s*b	(Conjugate transpose = Transpose)

DIAG    (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular.	= '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.

Page 1

DLATBS(3F)							    DLATBS(3F)

KD	     (input) INTEGER
The number	of subdiagonals	or superdiagonals in the triangular
matrix A.	KD >= 0.

AB	     (input) DOUBLE PRECISION 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 jth
column of the array AB as follows:  if UPLO = 'U', AB(kd+1+ij,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) DOUBLE PRECISION 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	 or
A'* 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 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 illegal value

FURTHER	DETAILS
A rough bound on x	is computed; if	that is	less than overflow, DTBSV is
called, otherwise,	specific code is used which checks for possible
overflow or divide-by-zero	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

Page 2

DLATBS(3F)							    DLATBS(3F)

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 DTBSV 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'*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 DTBSV if 1/M(n) and	1/G(n) are both	greater	than
max(underflow, 1/overflow).

Page 3

DLATBS(3F)							    DLATBS(3F)

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