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man pages->IRIX man pages -> complib/slatrs (3)
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### Contents

```
SLATRS(3F)							    SLATRS(3F)

```

### NAME[Toc][Back]

```     SLATRS - solve one	of the triangular systems   A *x = s*b or A'*x = s*b
with scaling to prevent overflow
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SLATRS(	UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,	CNORM,
INFO )

CHARACTER	DIAG, NORMIN, TRANS, UPLO

INTEGER	INFO, LDA, N

REAL		SCALE

REAL		A( LDA,	* ), CNORM( * ), X( * )
```

### PURPOSE[Toc][Back]

```     SLATRS solves one of the triangular systems triangular matrix, 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 threshold.  If the unscaled problem will
not cause overflow, the Level 2 BLAS routine STRSV	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

SLATRS(3F)							    SLATRS(3F)

A	     (input) REAL array, dimension (LDA,N)
The triangular matrix A.  If UPLO = 'U', the leading n by n upper
triangular	part of	the array A contains the upper triangular
matrix, and the strictly lower triangular part of A is not
referenced.  If UPLO = 'L', the leading n by n lower triangular
part of the array A contains the lower triangular matrix, and the
strictly upper triangular part of A is not	referenced.  If	DIAG =
'U', the diagonal elements	of A are also not referenced and are
assumed to	be 1.

LDA     (input) INTEGER
The leading dimension of the array	A.  LDA	>= max (1,N).

X	     (input/output) REAL 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	 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) 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 illegal value

FURTHER	DETAILS
A rough bound on x	is computed; if	that is	less than overflow, STRSV 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

SLATRS(3F)							    SLATRS(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 STRSV 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 STRSV if 1/M(n) and	1/G(n) are both	greater	than
max(underflow, 1/overflow).

Page 3

SLATRS(3F)							    SLATRS(3F)
SLATRS(3F)							    SLATRS(3F)

```

### NAME[Toc][Back]

```     SLATRS - solve one	of the triangular systems   A *x = s*b or A'*x = s*b
with scaling to prevent overflow
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SLATRS(	UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,	CNORM,
INFO )

CHARACTER	DIAG, NORMIN, TRANS, UPLO

INTEGER	INFO, LDA, N

REAL		SCALE

REAL		A( LDA,	* ), CNORM( * ), X( * )
```

### PURPOSE[Toc][Back]

```     SLATRS solves one of the triangular systems triangular matrix, 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 threshold.  If the unscaled problem will
not cause overflow, the Level 2 BLAS routine STRSV	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

SLATRS(3F)							    SLATRS(3F)

A	     (input) REAL array, dimension (LDA,N)
The triangular matrix A.  If UPLO = 'U', the leading n by n upper
triangular	part of	the array A contains the upper triangular
matrix, and the strictly lower triangular part of A is not
referenced.  If UPLO = 'L', the leading n by n lower triangular
part of the array A contains the lower triangular matrix, and the
strictly upper triangular part of A is not	referenced.  If	DIAG =
'U', the diagonal elements	of A are also not referenced and are
assumed to	be 1.

LDA     (input) INTEGER
The leading dimension of the array	A.  LDA	>= max (1,N).

X	     (input/output) REAL 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	 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) 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 illegal value

FURTHER	DETAILS
A rough bound on x	is computed; if	that is	less than overflow, STRSV 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

SLATRS(3F)							    SLATRS(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 STRSV 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 STRSV if 1/M(n) and	1/G(n) are both	greater	than
max(underflow, 1/overflow).

Page 3

SLATRS(3F)							    SLATRS(3F)

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