You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
359 lines
11 KiB
359 lines
11 KiB
2 years ago
|
*> \brief \b SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
|
||
|
|
*
|
||
|
|
* =========== DOCUMENTATION ===========
|
||
|
|
*
|
||
|
|
* Online html documentation available at
|
||
|
|
* http://www.netlib.org/lapack/explore-html/
|
||
|
|
*
|
||
|
|
*> \htmlonly
|
||
|
|
*> Download SLANTB + dependencies
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slantb.f">
|
||
|
|
*> [TGZ]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slantb.f">
|
||
|
|
*> [ZIP]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slantb.f">
|
||
|
|
*> [TXT]</a>
|
||
|
|
*> \endhtmlonly
|
||
|
|
*
|
||
|
|
* Definition:
|
||
|
|
* ===========
|
||
|
|
*
|
||
|
|
* REAL FUNCTION SLANTB( NORM, UPLO, DIAG, N, K, AB,
|
||
|
|
* LDAB, WORK )
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
* CHARACTER DIAG, NORM, UPLO
|
||
|
|
* INTEGER K, LDAB, N
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
* REAL AB( LDAB, * ), WORK( * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
*
|
||
|
|
*> \par Purpose:
|
||
|
|
* =============
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> SLANTB returns the value of the one norm, or the Frobenius norm, or
|
||
|
|
*> the infinity norm, or the element of largest absolute value of an
|
||
|
|
*> n by n triangular band matrix A, with ( k + 1 ) diagonals.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \return SLANTB
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> SLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
|
||
|
|
*> (
|
||
|
|
*> ( norm1(A), NORM = '1', 'O' or 'o'
|
||
|
|
*> (
|
||
|
|
*> ( normI(A), NORM = 'I' or 'i'
|
||
|
|
*> (
|
||
|
|
*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
|
||
|
|
*>
|
||
|
|
*> where norm1 denotes the one norm of a matrix (maximum column sum),
|
||
|
|
*> normI denotes the infinity norm of a matrix (maximum row sum) and
|
||
|
|
*> normF denotes the Frobenius norm of a matrix (square root of sum of
|
||
|
|
*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Arguments:
|
||
|
|
* ==========
|
||
|
|
*
|
||
|
|
*> \param[in] NORM
|
||
|
|
*> \verbatim
|
||
|
|
*> NORM is CHARACTER*1
|
||
|
|
*> Specifies the value to be returned in SLANTB as described
|
||
|
|
*> above.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] UPLO
|
||
|
|
*> \verbatim
|
||
|
|
*> UPLO is CHARACTER*1
|
||
|
|
*> Specifies whether the matrix A is upper or lower triangular.
|
||
|
|
*> = 'U': Upper triangular
|
||
|
|
*> = 'L': Lower triangular
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] DIAG
|
||
|
|
*> \verbatim
|
||
|
|
*> DIAG is CHARACTER*1
|
||
|
|
*> Specifies whether or not the matrix A is unit triangular.
|
||
|
|
*> = 'N': Non-unit triangular
|
||
|
|
*> = 'U': Unit triangular
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] N
|
||
|
|
*> \verbatim
|
||
|
|
*> N is INTEGER
|
||
|
|
*> The order of the matrix A. N >= 0. When N = 0, SLANTB is
|
||
|
|
*> set to zero.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] K
|
||
|
|
*> \verbatim
|
||
|
|
*> K is INTEGER
|
||
|
|
*> The number of super-diagonals of the matrix A if UPLO = 'U',
|
||
|
|
*> or the number of sub-diagonals of the matrix A if UPLO = 'L'.
|
||
|
|
*> K >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] AB
|
||
|
|
*> \verbatim
|
||
|
|
*> AB is REAL array, dimension (LDAB,N)
|
||
|
|
*> The upper or lower triangular band matrix A, stored in the
|
||
|
|
*> first k+1 rows of AB. The j-th column of A is stored
|
||
|
|
*> in the j-th column of the array AB as follows:
|
||
|
|
*> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
|
||
|
|
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
|
||
|
|
*> Note that when DIAG = 'U', the elements of the array AB
|
||
|
|
*> corresponding to the diagonal elements of the matrix A are
|
||
|
|
*> not referenced, but are assumed to be one.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDAB
|
||
|
|
*> \verbatim
|
||
|
|
*> LDAB is INTEGER
|
||
|
|
*> The leading dimension of the array AB. LDAB >= K+1.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] WORK
|
||
|
|
*> \verbatim
|
||
|
|
*> WORK is REAL array, dimension (MAX(1,LWORK)),
|
||
|
|
*> where LWORK >= N when NORM = 'I'; otherwise, WORK is not
|
||
|
|
*> referenced.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Authors:
|
||
|
|
* ========
|
||
|
|
*
|
||
|
|
*> \author Univ. of Tennessee
|
||
|
|
*> \author Univ. of California Berkeley
|
||
|
|
*> \author Univ. of Colorado Denver
|
||
|
|
*> \author NAG Ltd.
|
||
|
|
*
|
||
|
|
*> \ingroup realOTHERauxiliary
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
REAL FUNCTION SLANTB( NORM, UPLO, DIAG, N, K, AB,
|
||
|
|
$ LDAB, WORK )
|
||
|
|
*
|
||
|
|
* -- LAPACK auxiliary routine --
|
||
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
CHARACTER DIAG, NORM, UPLO
|
||
|
|
INTEGER K, LDAB, N
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
REAL AB( LDAB, * ), WORK( * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
*
|
||
|
|
* .. Parameters ..
|
||
|
|
REAL ONE, ZERO
|
||
|
|
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
|
||
|
|
* ..
|
||
|
|
* .. Local Scalars ..
|
||
|
|
LOGICAL UDIAG
|
||
|
|
INTEGER I, J, L
|
||
|
|
REAL SCALE, SUM, VALUE
|
||
|
|
* ..
|
||
|
|
* .. External Subroutines ..
|
||
|
|
EXTERNAL SLASSQ
|
||
|
|
* ..
|
||
|
|
* .. External Functions ..
|
||
|
|
LOGICAL LSAME, SISNAN
|
||
|
|
EXTERNAL LSAME, SISNAN
|
||
|
|
* ..
|
||
|
|
* .. Intrinsic Functions ..
|
||
|
|
INTRINSIC ABS, MAX, MIN, SQRT
|
||
|
|
* ..
|
||
|
|
* .. Executable Statements ..
|
||
|
|
*
|
||
|
|
IF( N.EQ.0 ) THEN
|
||
|
|
VALUE = ZERO
|
||
|
|
ELSE IF( LSAME( NORM, 'M' ) ) THEN
|
||
|
|
*
|
||
|
|
* Find max(abs(A(i,j))).
|
||
|
|
*
|
||
|
|
IF( LSAME( DIAG, 'U' ) ) THEN
|
||
|
|
VALUE = ONE
|
||
|
|
IF( LSAME( UPLO, 'U' ) ) THEN
|
||
|
|
DO 20 J = 1, N
|
||
|
|
DO 10 I = MAX( K+2-J, 1 ), K
|
||
|
|
SUM = ABS( AB( I, J ) )
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
10 CONTINUE
|
||
|
|
20 CONTINUE
|
||
|
|
ELSE
|
||
|
|
DO 40 J = 1, N
|
||
|
|
DO 30 I = 2, MIN( N+1-J, K+1 )
|
||
|
|
SUM = ABS( AB( I, J ) )
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
30 CONTINUE
|
||
|
|
40 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
VALUE = ZERO
|
||
|
|
IF( LSAME( UPLO, 'U' ) ) THEN
|
||
|
|
DO 60 J = 1, N
|
||
|
|
DO 50 I = MAX( K+2-J, 1 ), K + 1
|
||
|
|
SUM = ABS( AB( I, J ) )
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
50 CONTINUE
|
||
|
|
60 CONTINUE
|
||
|
|
ELSE
|
||
|
|
DO 80 J = 1, N
|
||
|
|
DO 70 I = 1, MIN( N+1-J, K+1 )
|
||
|
|
SUM = ABS( AB( I, J ) )
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
70 CONTINUE
|
||
|
|
80 CONTINUE
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
|
||
|
|
*
|
||
|
|
* Find norm1(A).
|
||
|
|
*
|
||
|
|
VALUE = ZERO
|
||
|
|
UDIAG = LSAME( DIAG, 'U' )
|
||
|
|
IF( LSAME( UPLO, 'U' ) ) THEN
|
||
|
|
DO 110 J = 1, N
|
||
|
|
IF( UDIAG ) THEN
|
||
|
|
SUM = ONE
|
||
|
|
DO 90 I = MAX( K+2-J, 1 ), K
|
||
|
|
SUM = SUM + ABS( AB( I, J ) )
|
||
|
|
90 CONTINUE
|
||
|
|
ELSE
|
||
|
|
SUM = ZERO
|
||
|
|
DO 100 I = MAX( K+2-J, 1 ), K + 1
|
||
|
|
SUM = SUM + ABS( AB( I, J ) )
|
||
|
|
100 CONTINUE
|
||
|
|
END IF
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
110 CONTINUE
|
||
|
|
ELSE
|
||
|
|
DO 140 J = 1, N
|
||
|
|
IF( UDIAG ) THEN
|
||
|
|
SUM = ONE
|
||
|
|
DO 120 I = 2, MIN( N+1-J, K+1 )
|
||
|
|
SUM = SUM + ABS( AB( I, J ) )
|
||
|
|
120 CONTINUE
|
||
|
|
ELSE
|
||
|
|
SUM = ZERO
|
||
|
|
DO 130 I = 1, MIN( N+1-J, K+1 )
|
||
|
|
SUM = SUM + ABS( AB( I, J ) )
|
||
|
|
130 CONTINUE
|
||
|
|
END IF
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
140 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE IF( LSAME( NORM, 'I' ) ) THEN
|
||
|
|
*
|
||
|
|
* Find normI(A).
|
||
|
|
*
|
||
|
|
VALUE = ZERO
|
||
|
|
IF( LSAME( UPLO, 'U' ) ) THEN
|
||
|
|
IF( LSAME( DIAG, 'U' ) ) THEN
|
||
|
|
DO 150 I = 1, N
|
||
|
|
WORK( I ) = ONE
|
||
|
|
150 CONTINUE
|
||
|
|
DO 170 J = 1, N
|
||
|
|
L = K + 1 - J
|
||
|
|
DO 160 I = MAX( 1, J-K ), J - 1
|
||
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
|
||
|
|
160 CONTINUE
|
||
|
|
170 CONTINUE
|
||
|
|
ELSE
|
||
|
|
DO 180 I = 1, N
|
||
|
|
WORK( I ) = ZERO
|
||
|
|
180 CONTINUE
|
||
|
|
DO 200 J = 1, N
|
||
|
|
L = K + 1 - J
|
||
|
|
DO 190 I = MAX( 1, J-K ), J
|
||
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
|
||
|
|
190 CONTINUE
|
||
|
|
200 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
IF( LSAME( DIAG, 'U' ) ) THEN
|
||
|
|
DO 210 I = 1, N
|
||
|
|
WORK( I ) = ONE
|
||
|
|
210 CONTINUE
|
||
|
|
DO 230 J = 1, N
|
||
|
|
L = 1 - J
|
||
|
|
DO 220 I = J + 1, MIN( N, J+K )
|
||
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
|
||
|
|
220 CONTINUE
|
||
|
|
230 CONTINUE
|
||
|
|
ELSE
|
||
|
|
DO 240 I = 1, N
|
||
|
|
WORK( I ) = ZERO
|
||
|
|
240 CONTINUE
|
||
|
|
DO 260 J = 1, N
|
||
|
|
L = 1 - J
|
||
|
|
DO 250 I = J, MIN( N, J+K )
|
||
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
|
||
|
|
250 CONTINUE
|
||
|
|
260 CONTINUE
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
DO 270 I = 1, N
|
||
|
|
SUM = WORK( I )
|
||
|
|
IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
|
||
|
|
270 CONTINUE
|
||
|
|
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
|
||
|
|
*
|
||
|
|
* Find normF(A).
|
||
|
|
*
|
||
|
|
IF( LSAME( UPLO, 'U' ) ) THEN
|
||
|
|
IF( LSAME( DIAG, 'U' ) ) THEN
|
||
|
|
SCALE = ONE
|
||
|
|
SUM = N
|
||
|
|
IF( K.GT.0 ) THEN
|
||
|
|
DO 280 J = 2, N
|
||
|
|
CALL SLASSQ( MIN( J-1, K ),
|
||
|
|
$ AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
|
||
|
|
$ SUM )
|
||
|
|
280 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
SCALE = ZERO
|
||
|
|
SUM = ONE
|
||
|
|
DO 290 J = 1, N
|
||
|
|
CALL SLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ),
|
||
|
|
$ 1, SCALE, SUM )
|
||
|
|
290 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
IF( LSAME( DIAG, 'U' ) ) THEN
|
||
|
|
SCALE = ONE
|
||
|
|
SUM = N
|
||
|
|
IF( K.GT.0 ) THEN
|
||
|
|
DO 300 J = 1, N - 1
|
||
|
|
CALL SLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
|
||
|
|
$ SUM )
|
||
|
|
300 CONTINUE
|
||
|
|
END IF
|
||
|
|
ELSE
|
||
|
|
SCALE = ZERO
|
||
|
|
SUM = ONE
|
||
|
|
DO 310 J = 1, N
|
||
|
|
CALL SLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE,
|
||
|
|
$ SUM )
|
||
|
|
310 CONTINUE
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
VALUE = SCALE*SQRT( SUM )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
SLANTB = VALUE
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of SLANTB
|
||
|
|
*
|
||
|
|
END
|