#include "blaswrap.h"
#include "f2c.h"
/* Subroutine */ int slatrs_(char *uplo, char *trans, char *diag, char *
normin, integer *n, real *a, integer *lda, real *x, real *scale, real
*cnorm, integer *info )
{
/* -- LAPACK auxiliary routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
SLATRS solves one of the triangular systems
A *x = s*b or A'*x = s*b
with scaling to prevent overflow. Here A is an upper or lower
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
=========
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.
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
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).
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static real c_b36 = .5f;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
real r__1, r__2, r__3;
/* Local variables */
static integer i__, j;
static real xj, rec, tjj;
static integer jinc;
static real xbnd;
static integer imax;
static real tmax, tjjs;
extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
static real xmax, grow, sumj;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static real tscal, uscal;
static integer jlast;
extern doublereal sasum_(integer *, real *, integer *);
static logical upper;
extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
real *, integer *), strsv_(char *, char *, char *, integer *,
real *, integer *, real *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
static real bignum;
extern integer isamax_(integer *, real *, integer *);
static logical notran;
static integer jfirst;
static real smlnum;
static logical nounit;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--x;
--cnorm;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
/* Test the input parameters. */
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (! lsame_(normin, "Y") && ! lsame_(normin,
"N")) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLATRS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Determine machine dependent parameters to control overflow. */
smlnum = slamch_("Safe minimum") / slamch_("Precision");
bignum = 1.f / smlnum;
*scale = 1.f;
if (lsame_(normin, "N")) {
/* Compute the 1-norm of each column, not including the diagonal. */
if (upper) {
/* A is upper triangular. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
cnorm[j] = sasum_(&i__2, &a[j * a_dim1 + 1], &c__1);
/* L10: */
}
} else {
/* A is lower triangular. */
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
i__2 = *n - j;
cnorm[j] = sasum_(&i__2, &a[j + 1 + j * a_dim1], &c__1);
/* L20: */
}
cnorm[*n] = 0.f;
}
}
/* Scale the column norms by TSCAL if the maximum element in CNORM is
greater than BIGNUM. */
imax = isamax_(n, &cnorm[1], &c__1);
tmax = cnorm[imax];
if (tmax <= bignum) {
tscal = 1.f;
} else {
tscal = 1.f / (smlnum * tmax);
sscal_(n, &tscal, &cnorm[1], &c__1);
}
/* Compute a bound on the computed solution vector to see if the
Level 2 BLAS routine STRSV can be used. */
j = isamax_(n, &x[1], &c__1);
xmax = (r__1 = x[j], dabs(r__1));
xbnd = xmax;
if (notran) {
/* Compute the growth in A * x = b. */
if (upper) {
jfirst = *n;
jlast = 1;
jinc = -1;
} else {
jfirst = 1;
jlast = *n;
jinc = 1;
}
if (tscal != 1.f) {
grow = 0.f;
goto L50;
}
if (nounit) {
/* A is non-unit triangular.
Compute GROW = 1/G(j) and XBND = 1/M(j).
Initially, G(0) = max{x(i), i=1,...,n}. */
grow = 1.f / dmax(xbnd,smlnum);
xbnd = grow;
i__1 = jlast;
i__2 = jinc;
for (j = jfirst; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Exit the loop if the growth factor is too small. */
if (grow <= smlnum) {
goto L50;
}
/* M(j) = G(j-1) / abs(A(j,j)) */
tjj = (r__1 = a[j + j * a_dim1], dabs(r__1));
/* Computing MIN */
r__1 = xbnd, r__2 = dmin(1.f,tjj) * grow;
xbnd = dmin(r__1,r__2);
if (tjj + cnorm[j] >= smlnum) {
/* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) */
grow *= tjj / (tjj + cnorm[j]);
} else {
/* G(j) could overflow, set GROW to 0. */
grow = 0.f;
}
/* L30: */
}
grow = xbnd;
} else {
/* A is unit triangular.
Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
Computing MIN */
r__1 = 1.f, r__2 = 1.f / dmax(xbnd,smlnum);
grow = dmin(r__1,r__2);
i__2 = jlast;
i__1 = jinc;
for (j = jfirst; i__1 < 0 ? j >= i__2 : j <= i__2; j += i__1) {
/* Exit the loop if the growth factor is too small. */
if (grow <= smlnum) {
goto L50;
}
/* G(j) = G(j-1)*( 1 + CNORM(j) ) */
grow *= 1.f / (cnorm[j] + 1.f);
/* L40: */
}
}
L50:
;
} else {
/* Compute the growth in A' * x = b. */
if (upper) {
jfirst = 1;
jlast = *n;
jinc = 1;
} else {
jfirst = *n;
jlast = 1;
jinc = -1;
}
if (tscal != 1.f) {
grow = 0.f;
goto L80;
}
if (nounit) {
/* A is non-unit triangular.
Compute GROW = 1/G(j) and XBND = 1/M(j).
Initially, M(0) = max{x(i), i=1,...,n}. */
grow = 1.f / dmax(xbnd,smlnum);
xbnd = grow;
i__1 = jlast;
i__2 = jinc;
for (j = jfirst; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Exit the loop if the growth factor is too small. */
if (grow <= smlnum) {
goto L80;
}
/* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) */
xj = cnorm[j] + 1.f;
/* Computing MIN */
r__1 = grow, r__2 = xbnd / xj;
grow = dmin(r__1,r__2);
/* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) */
tjj = (r__1 = a[j + j * a_dim1], dabs(r__1));
if (xj > tjj) {
xbnd *= tjj / xj;
}
/* L60: */
}
grow = dmin(grow,xbnd);
} else {
/* A is unit triangular.
Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
Computing MIN */
r__1 = 1.f, r__2 = 1.f / dmax(xbnd,smlnum);
grow = dmin(r__1,r__2);
i__2 = jlast;
i__1 = jinc;
for (j = jfirst; i__1 < 0 ? j >= i__2 : j <= i__2; j += i__1) {
/* Exit the loop if the growth factor is too small. */
if (grow <= smlnum) {
goto L80;
}
/* G(j) = ( 1 + CNORM(j) )*G(j-1) */
xj = cnorm[j] + 1.f;
grow /= xj;
/* L70: */
}
}
L80:
;
}
if (grow * tscal > smlnum) {
/* Use the Level 2 BLAS solve if the reciprocal of the bound on
elements of X is not too small. */
strsv_(uplo, trans, diag, n, &a[a_offset], lda, &x[1], &c__1);
} else {
/* Use a Level 1 BLAS solve, scaling intermediate results. */
if (xmax > bignum) {
/* Scale X so that its components are less than or equal to
BIGNUM in absolute value. */
*scale = bignum / xmax;
sscal_(n, scale, &x[1], &c__1);
xmax = bignum;
}
if (notran) {
/* Solve A * x = b */
i__1 = jlast;
i__2 = jinc;
for (j = jfirst; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Compute x(j) = b(j) / A(j,j), scaling x if necessary. */
xj = (r__1 = x[j], dabs(r__1));
if (nounit) {
tjjs = a[j + j * a_dim1] * tscal;
} else {
tjjs = tscal;
if (tscal == 1.f) {
goto L95;
}
}
tjj = dabs(tjjs);
if (tjj > smlnum) {
/* abs(A(j,j)) > SMLNUM: */
if (tjj < 1.f) {
if (xj > tjj * bignum) {
/* Scale x by 1/b(j). */
rec = 1.f / xj;
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
xmax *= rec;
}
}
x[j] /= tjjs;
xj = (r__1 = x[j], dabs(r__1));
} else if (tjj > 0.f) {
/* 0 < abs(A(j,j)) <= SMLNUM: */
if (xj > tjj * bignum) {
/* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
to avoid overflow when dividing by A(j,j). */
rec = tjj * bignum / xj;
if (cnorm[j] > 1.f) {
/* Scale by 1/CNORM(j) to avoid overflow when
multiplying x(j) times column j. */
rec /= cnorm[j];
}
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
xmax *= rec;
}
x[j] /= tjjs;
xj = (r__1 = x[j], dabs(r__1));
} else {
/* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
scale = 0, and compute a solution to A*x = 0. */
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
x[i__] = 0.f;
/* L90: */
}
x[j] = 1.f;
xj = 1.f;
*scale = 0.f;
xmax = 0.f;
}
L95:
/* Scale x if necessary to avoid overflow when adding a
multiple of column j of A. */
if (xj > 1.f) {
rec = 1.f / xj;
if (cnorm[j] > (bignum - xmax) * rec) {
/* Scale x by 1/(2*abs(x(j))). */
rec *= .5f;
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
}
} else if (xj * cnorm[j] > bignum - xmax) {
/* Scale x by 1/2. */
sscal_(n, &c_b36, &x[1], &c__1);
*scale *= .5f;
}
if (upper) {
if (j > 1) {
/* Compute the update
x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j) */
i__3 = j - 1;
r__1 = -x[j] * tscal;
saxpy_(&i__3, &r__1, &a[j * a_dim1 + 1], &c__1, &x[1],
&c__1);
i__3 = j - 1;
i__ = isamax_(&i__3, &x[1], &c__1);
xmax = (r__1 = x[i__], dabs(r__1));
}
} else {
if (j < *n) {
/* Compute the update
x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j) */
i__3 = *n - j;
r__1 = -x[j] * tscal;
saxpy_(&i__3, &r__1, &a[j + 1 + j * a_dim1], &c__1, &
x[j + 1], &c__1);
i__3 = *n - j;
i__ = j + isamax_(&i__3, &x[j + 1], &c__1);
xmax = (r__1 = x[i__], dabs(r__1));
}
}
/* L100: */
}
} else {
/* Solve A' * x = b */
i__2 = jlast;
i__1 = jinc;
for (j = jfirst; i__1 < 0 ? j >= i__2 : j <= i__2; j += i__1) {
/* Compute x(j) = b(j) - sum A(k,j)*x(k).
k<>j */
xj = (r__1 = x[j], dabs(r__1));
uscal = tscal;
rec = 1.f / dmax(xmax,1.f);
if (cnorm[j] > (bignum - xj) * rec) {
/* If x(j) could overflow, scale x by 1/(2*XMAX). */
rec *= .5f;
if (nounit) {
tjjs = a[j + j * a_dim1] * tscal;
} else {
tjjs = tscal;
}
tjj = dabs(tjjs);
if (tjj > 1.f) {
/* Divide by A(j,j) when scaling x if A(j,j) > 1.
Computing MIN */
r__1 = 1.f, r__2 = rec * tjj;
rec = dmin(r__1,r__2);
uscal /= tjjs;
}
if (rec < 1.f) {
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
xmax *= rec;
}
}
sumj = 0.f;
if (uscal == 1.f) {
/* If the scaling needed for A in the dot product is 1,
call SDOT to perform the dot product. */
if (upper) {
i__3 = j - 1;
sumj = sdot_(&i__3, &a[j * a_dim1 + 1], &c__1, &x[1],
&c__1);
} else if (j < *n) {
i__3 = *n - j;
sumj = sdot_(&i__3, &a[j + 1 + j * a_dim1], &c__1, &x[
j + 1], &c__1);
}
} else {
/* Otherwise, use in-line code for the dot product. */
if (upper) {
i__3 = j - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
sumj += a[i__ + j * a_dim1] * uscal * x[i__];
/* L110: */
}
} else if (j < *n) {
i__3 = *n;
for (i__ = j + 1; i__ <= i__3; ++i__) {
sumj += a[i__ + j * a_dim1] * uscal * x[i__];
/* L120: */
}
}
}
if (uscal == tscal) {
/* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
was not used to scale the dotproduct. */
x[j] -= sumj;
xj = (r__1 = x[j], dabs(r__1));
if (nounit) {
tjjs = a[j + j * a_dim1] * tscal;
} else {
tjjs = tscal;
if (tscal == 1.f) {
goto L135;
}
}
/* Compute x(j) = x(j) / A(j,j), scaling if necessary. */
tjj = dabs(tjjs);
if (tjj > smlnum) {
/* abs(A(j,j)) > SMLNUM: */
if (tjj < 1.f) {
if (xj > tjj * bignum) {
/* Scale X by 1/abs(x(j)). */
rec = 1.f / xj;
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
xmax *= rec;
}
}
x[j] /= tjjs;
} else if (tjj > 0.f) {
/* 0 < abs(A(j,j)) <= SMLNUM: */
if (xj > tjj * bignum) {
/* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. */
rec = tjj * bignum / xj;
sscal_(n, &rec, &x[1], &c__1);
*scale *= rec;
xmax *= rec;
}
x[j] /= tjjs;
} else {
/* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
scale = 0, and compute a solution to A'*x = 0. */
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
x[i__] = 0.f;
/* L130: */
}
x[j] = 1.f;
*scale = 0.f;
xmax = 0.f;
}
L135:
;
} else {
/* Compute x(j) := x(j) / A(j,j) - sumj if the dot
product has already been divided by 1/A(j,j). */
x[j] = x[j] / tjjs - sumj;
}
/* Computing MAX */
r__2 = xmax, r__3 = (r__1 = x[j], dabs(r__1));
xmax = dmax(r__2,r__3);
/* L140: */
}
}
*scale /= tscal;
}
/* Scale the column norms by 1/TSCAL for return. */
if (tscal != 1.f) {
r__1 = 1.f / tscal;
sscal_(n, &r__1, &cnorm[1], &c__1);
}
return 0;
/* End of SLATRS */
} /* slatrs_ */