DOUBLE PRECISION FUNCTION ZLANHF( NORM, TRANSR, UPLO, N, A, WORK ) * * -- LAPACK routine (version 3.2.1) -- * * -- Contributed by Fred Gustavson of the IBM Watson Research Center -- * -- April 2009 -- * * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER NORM, TRANSR, UPLO INTEGER N * .. * .. Array Arguments .. DOUBLE PRECISION WORK( 0: * ) COMPLEX*16 A( 0: * ) * .. * * Purpose * ======= * * ZLANHF returns the value of the one norm, or the Frobenius norm, or * the infinity norm, or the element of largest absolute value of a * complex Hermitian matrix A in RFP format. * * Description * =========== * * ZLANHF returns the value * * ZLANHF = ( 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 matrix norm. * * Arguments * ========= * * NORM (input) CHARACTER * Specifies the value to be returned in ZLANHF as described * above. * * TRANSR (input) CHARACTER * Specifies whether the RFP format of A is normal or * conjugate-transposed format. * = 'N': RFP format is Normal * = 'C': RFP format is Conjugate-transposed * * UPLO (input) CHARACTER * On entry, UPLO specifies whether the RFP matrix A came from * an upper or lower triangular matrix as follows: * * UPLO = 'U' or 'u' RFP A came from an upper triangular * matrix * * UPLO = 'L' or 'l' RFP A came from a lower triangular * matrix * * N (input) INTEGER * The order of the matrix A. N >= 0. When N = 0, ZLANHF is * set to zero. * * A (input) COMPLEX*16 array, dimension ( N*(N+1)/2 ); * On entry, the matrix A in RFP Format. * RFP Format is described by TRANSR, UPLO and N as follows: * If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even; * K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If * TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A * as defined when TRANSR = 'N'. The contents of RFP A are * defined by UPLO as follows: If UPLO = 'U' the RFP A * contains the ( N*(N+1)/2 ) elements of upper packed A * either in normal or conjugate-transpose Format. If * UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements * of lower packed A either in normal or conjugate-transpose * Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When * TRANSR is 'N' the LDA is N+1 when N is even and is N when * is odd. See the Note below for more details. * Unchanged on exit. * * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), * where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, * WORK is not referenced. * * Further Details * =============== * * We first consider Standard Packed Format when N is even. * We give an example where N = 6. * * AP is Upper AP is Lower * * 00 01 02 03 04 05 00 * 11 12 13 14 15 10 11 * 22 23 24 25 20 21 22 * 33 34 35 30 31 32 33 * 44 45 40 41 42 43 44 * 55 50 51 52 53 54 55 * * * Let TRANSR = 'N'. RFP holds AP as follows: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last * three columns of AP upper. The lower triangle A(4:6,0:2) consists of * conjugate-transpose of the first three columns of AP upper. * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:2,0:2) consists of * conjugate-transpose of the last three columns of AP lower. * To denote conjugate we place -- above the element. This covers the * case N even and TRANSR = 'N'. * * RFP A RFP A * * -- -- -- * 03 04 05 33 43 53 * -- -- * 13 14 15 00 44 54 * -- * 23 24 25 10 11 55 * * 33 34 35 20 21 22 * -- * 00 44 45 30 31 32 * -- -- * 01 11 55 40 41 42 * -- -- -- * 02 12 22 50 51 52 * * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * * RFP A RFP A * * -- -- -- -- -- -- -- -- -- -- * 03 13 23 33 00 01 02 33 00 10 20 30 40 50 * -- -- -- -- -- -- -- -- -- -- * 04 14 24 34 44 11 12 43 44 11 21 31 41 51 * -- -- -- -- -- -- -- -- -- -- * 05 15 25 35 45 55 22 53 54 55 22 32 42 52 * * * We next consider Standard Packed Format when N is odd. * We give an example where N = 5. * * AP is Upper AP is Lower * * 00 01 02 03 04 00 * 11 12 13 14 10 11 * 22 23 24 20 21 22 * 33 34 30 31 32 33 * 44 40 41 42 43 44 * * * Let TRANSR = 'N'. RFP holds AP as follows: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last * three columns of AP upper. The lower triangle A(3:4,0:1) consists of * conjugate-transpose of the first two columns of AP upper. * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:1,1:2) consists of * conjugate-transpose of the last two columns of AP lower. * To denote conjugate we place -- above the element. This covers the * case N odd and TRANSR = 'N'. * * RFP A RFP A * * -- -- * 02 03 04 00 33 43 * -- * 12 13 14 10 11 44 * * 22 23 24 20 21 22 * -- * 00 33 34 30 31 32 * -- -- * 01 11 44 40 41 42 * * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * * RFP A RFP A * * -- -- -- -- -- -- -- -- -- * 02 12 22 00 01 00 10 20 30 40 50 * -- -- -- -- -- -- -- -- -- * 03 13 23 33 11 33 11 21 31 41 51 * -- -- -- -- -- -- -- -- -- * 04 14 24 34 44 43 44 22 32 42 52 * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER I, J, IFM, ILU, NOE, N1, K, L, LDA DOUBLE PRECISION SCALE, S, VALUE, AA * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX EXTERNAL LSAME, IDAMAX * .. * .. External Subroutines .. EXTERNAL ZLASSQ * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, SQRT * .. * .. Executable Statements .. * IF( N.EQ.0 ) THEN ZLANHF = ZERO RETURN END IF * * set noe = 1 if n is odd. if n is even set noe=0 * NOE = 1 IF( MOD( N, 2 ).EQ.0 ) + NOE = 0 * * set ifm = 0 when form='C' or 'c' and 1 otherwise * IFM = 1 IF( LSAME( TRANSR, 'C' ) ) + IFM = 0 * * set ilu = 0 when uplo='U or 'u' and 1 otherwise * ILU = 1 IF( LSAME( UPLO, 'U' ) ) + ILU = 0 * * set lda = (n+1)/2 when ifm = 0 * set lda = n when ifm = 1 and noe = 1 * set lda = n+1 when ifm = 1 and noe = 0 * IF( IFM.EQ.1 ) THEN IF( NOE.EQ.1 ) THEN LDA = N ELSE * noe=0 LDA = N + 1 END IF ELSE * ifm=0 LDA = ( N+1 ) / 2 END IF * IF( LSAME( NORM, 'M' ) ) THEN * * Find max(abs(A(i,j))). * K = ( N+1 ) / 2 VALUE = ZERO IF( NOE.EQ.1 ) THEN * n is odd & n = k + k - 1 IF( IFM.EQ.1 ) THEN * A is n by k IF( ILU.EQ.1 ) THEN * uplo ='L' J = 0 * -> L(0,0) VALUE = MAX( VALUE, ABS( DBLE( A( J+J*LDA ) ) ) ) DO I = 1, N - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO DO J = 1, K - 1 DO I = 0, J - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J - 1 * L(k+j,k+j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J * -> L(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J + 1, N - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO ELSE * uplo = 'U' DO J = 0, K - 2 DO I = 0, K + J - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = K + J - 1 * -> U(i,i) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = I + 1 * =k+j; i -> U(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = K + J + 1, N - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO DO I = 0, N - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) * j=k-1 END DO * i=n-1 -> U(n-1,n-1) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) END IF ELSE * xpose case; A is k by n IF( ILU.EQ.1 ) THEN * uplo ='L' DO J = 0, K - 2 DO I = 0, J - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J * L(i,i) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J + 1 * L(j+k,j+k) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J + 2, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO J = K - 1 DO I = 0, K - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = K - 1 * -> L(i,i) is at A(i,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO J = K, N - 1 DO I = 0, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO ELSE * uplo = 'U' DO J = 0, K - 2 DO I = 0, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO J = K - 1 * -> U(j,j) is at A(0,j) VALUE = MAX( VALUE, ABS( DBLE( A( 0+J*LDA ) ) ) ) DO I = 1, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO DO J = K, N - 1 DO I = 0, J - K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J - K * -> U(i,i) at A(i,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J - K + 1 * U(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J - K + 2, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO END IF END IF ELSE * n is even & k = n/2 IF( IFM.EQ.1 ) THEN * A is n+1 by k IF( ILU.EQ.1 ) THEN * uplo ='L' J = 0 * -> L(k,k) & j=1 -> L(0,0) VALUE = MAX( VALUE, ABS( DBLE( A( J+J*LDA ) ) ) ) VALUE = MAX( VALUE, ABS( DBLE( A( J+1+J*LDA ) ) ) ) DO I = 2, N VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO DO J = 1, K - 1 DO I = 0, J - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J * L(k+j,k+j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J + 1 * -> L(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J + 2, N VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO ELSE * uplo = 'U' DO J = 0, K - 2 DO I = 0, K + J - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = K + J * -> U(i,i) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = I + 1 * =k+j+1; i -> U(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = K + J + 2, N VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO DO I = 0, N - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) * j=k-1 END DO * i=n-1 -> U(n-1,n-1) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = N * -> U(k-1,k-1) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) END IF ELSE * xpose case; A is k by n+1 IF( ILU.EQ.1 ) THEN * uplo ='L' J = 0 * -> L(k,k) at A(0,0) VALUE = MAX( VALUE, ABS( DBLE( A( J+J*LDA ) ) ) ) DO I = 1, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO DO J = 1, K - 1 DO I = 0, J - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J - 1 * L(i,i) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J * L(j+k,j+k) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J + 1, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO J = K DO I = 0, K - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = K - 1 * -> L(i,i) is at A(i,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO J = K + 1, N DO I = 0, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO ELSE * uplo = 'U' DO J = 0, K - 1 DO I = 0, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO J = K * -> U(j,j) is at A(0,j) VALUE = MAX( VALUE, ABS( DBLE( A( 0+J*LDA ) ) ) ) DO I = 1, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO DO J = K + 1, N - 1 DO I = 0, J - K - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = J - K - 1 * -> U(i,i) at A(i,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) I = J - K * U(j,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) DO I = J - K + 1, K - 1 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO END DO J = N DO I = 0, K - 2 VALUE = MAX( VALUE, ABS( A( I+J*LDA ) ) ) END DO I = K - 1 * U(k,k) at A(i,j) VALUE = MAX( VALUE, ABS( DBLE( A( I+J*LDA ) ) ) ) END IF END IF END IF ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR. + ( NORM.EQ.'1' ) ) THEN * * Find normI(A) ( = norm1(A), since A is Hermitian). * IF( IFM.EQ.1 ) THEN * A is 'N' K = N / 2 IF( NOE.EQ.1 ) THEN * n is odd & A is n by (n+1)/2 IF( ILU.EQ.0 ) THEN * uplo = 'U' DO I = 0, K - 1 WORK( I ) = ZERO END DO DO J = 0, K S = ZERO DO I = 0, K + J - 1 AA = ABS( A( I+J*LDA ) ) * -> A(i,j+k) S = S + AA WORK( I ) = WORK( I ) + AA END DO AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j+k,j+k) WORK( J+K ) = S + AA IF( I.EQ.K+K ) + GO TO 10 I = I + 1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j,j) WORK( J ) = WORK( J ) + AA S = ZERO DO L = J + 1, K - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * -> A(l,j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( J ) = WORK( J ) + S END DO 10 CONTINUE I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) ELSE * ilu = 1 & uplo = 'L' K = K + 1 * k=(n+1)/2 for n odd and ilu=1 DO I = K, N - 1 WORK( I ) = ZERO END DO DO J = K - 1, 0, -1 S = ZERO DO I = 0, J - 2 AA = ABS( A( I+J*LDA ) ) * -> A(j+k,i+k) S = S + AA WORK( I+K ) = WORK( I+K ) + AA END DO IF( J.GT.0 ) THEN AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j+k,j+k) S = S + AA WORK( I+K ) = WORK( I+K ) + S * i=j I = I + 1 END IF AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j,j) WORK( J ) = AA S = ZERO DO L = J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * -> A(l,j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( J ) = WORK( J ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) END IF ELSE * n is even & A is n+1 by k = n/2 IF( ILU.EQ.0 ) THEN * uplo = 'U' DO I = 0, K - 1 WORK( I ) = ZERO END DO DO J = 0, K - 1 S = ZERO DO I = 0, K + J - 1 AA = ABS( A( I+J*LDA ) ) * -> A(i,j+k) S = S + AA WORK( I ) = WORK( I ) + AA END DO AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j+k,j+k) WORK( J+K ) = S + AA I = I + 1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j,j) WORK( J ) = WORK( J ) + AA S = ZERO DO L = J + 1, K - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * -> A(l,j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( J ) = WORK( J ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) ELSE * ilu = 1 & uplo = 'L' DO I = K, N - 1 WORK( I ) = ZERO END DO DO J = K - 1, 0, -1 S = ZERO DO I = 0, J - 1 AA = ABS( A( I+J*LDA ) ) * -> A(j+k,i+k) S = S + AA WORK( I+K ) = WORK( I+K ) + AA END DO AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j+k,j+k) S = S + AA WORK( I+K ) = WORK( I+K ) + S * i=j I = I + 1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * -> A(j,j) WORK( J ) = AA S = ZERO DO L = J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * -> A(l,j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( J ) = WORK( J ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) END IF END IF ELSE * ifm=0 K = N / 2 IF( NOE.EQ.1 ) THEN * n is odd & A is (n+1)/2 by n IF( ILU.EQ.0 ) THEN * uplo = 'U' N1 = K * n/2 K = K + 1 * k is the row size and lda DO I = N1, N - 1 WORK( I ) = ZERO END DO DO J = 0, N1 - 1 S = ZERO DO I = 0, K - 1 AA = ABS( A( I+J*LDA ) ) * A(j,n1+i) WORK( I+N1 ) = WORK( I+N1 ) + AA S = S + AA END DO WORK( J ) = S END DO * j=n1=k-1 is special S = ABS( DBLE( A( 0+J*LDA ) ) ) * A(k-1,k-1) DO I = 1, K - 1 AA = ABS( A( I+J*LDA ) ) * A(k-1,i+n1) WORK( I+N1 ) = WORK( I+N1 ) + AA S = S + AA END DO WORK( J ) = WORK( J ) + S DO J = K, N - 1 S = ZERO DO I = 0, J - K - 1 AA = ABS( A( I+J*LDA ) ) * A(i,j-k) WORK( I ) = WORK( I ) + AA S = S + AA END DO * i=j-k AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(j-k,j-k) S = S + AA WORK( J-K ) = WORK( J-K ) + S I = I + 1 S = ABS( DBLE( A( I+J*LDA ) ) ) * A(j,j) DO L = J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * A(j,l) WORK( L ) = WORK( L ) + AA S = S + AA END DO WORK( J ) = WORK( J ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) ELSE * ilu=1 & uplo = 'L' K = K + 1 * k=(n+1)/2 for n odd and ilu=1 DO I = K, N - 1 WORK( I ) = ZERO END DO DO J = 0, K - 2 * process S = ZERO DO I = 0, J - 1 AA = ABS( A( I+J*LDA ) ) * A(j,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO AA = ABS( DBLE( A( I+J*LDA ) ) ) * i=j so process of A(j,j) S = S + AA WORK( J ) = S * is initialised here I = I + 1 * i=j process A(j+k,j+k) AA = ABS( DBLE( A( I+J*LDA ) ) ) S = AA DO L = K + J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * A(l,k+j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( K+J ) = WORK( K+J ) + S END DO * j=k-1 is special :process col A(k-1,0:k-1) S = ZERO DO I = 0, K - 2 AA = ABS( A( I+J*LDA ) ) * A(k,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO * i=k-1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(k-1,k-1) S = S + AA WORK( I ) = S * done with col j=k+1 DO J = K, N - 1 * process col j of A = A(j,0:k-1) S = ZERO DO I = 0, K - 1 AA = ABS( A( I+J*LDA ) ) * A(j,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO WORK( J ) = WORK( J ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) END IF ELSE * n is even & A is k=n/2 by n+1 IF( ILU.EQ.0 ) THEN * uplo = 'U' DO I = K, N - 1 WORK( I ) = ZERO END DO DO J = 0, K - 1 S = ZERO DO I = 0, K - 1 AA = ABS( A( I+J*LDA ) ) * A(j,i+k) WORK( I+K ) = WORK( I+K ) + AA S = S + AA END DO WORK( J ) = S END DO * j=k AA = ABS( DBLE( A( 0+J*LDA ) ) ) * A(k,k) S = AA DO I = 1, K - 1 AA = ABS( A( I+J*LDA ) ) * A(k,k+i) WORK( I+K ) = WORK( I+K ) + AA S = S + AA END DO WORK( J ) = WORK( J ) + S DO J = K + 1, N - 1 S = ZERO DO I = 0, J - 2 - K AA = ABS( A( I+J*LDA ) ) * A(i,j-k-1) WORK( I ) = WORK( I ) + AA S = S + AA END DO * i=j-1-k AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(j-k-1,j-k-1) S = S + AA WORK( J-K-1 ) = WORK( J-K-1 ) + S I = I + 1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(j,j) S = AA DO L = J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * A(j,l) WORK( L ) = WORK( L ) + AA S = S + AA END DO WORK( J ) = WORK( J ) + S END DO * j=n S = ZERO DO I = 0, K - 2 AA = ABS( A( I+J*LDA ) ) * A(i,k-1) WORK( I ) = WORK( I ) + AA S = S + AA END DO * i=k-1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(k-1,k-1) S = S + AA WORK( I ) = WORK( I ) + S I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) ELSE * ilu=1 & uplo = 'L' DO I = K, N - 1 WORK( I ) = ZERO END DO * j=0 is special :process col A(k:n-1,k) S = ABS( DBLE( A( 0 ) ) ) * A(k,k) DO I = 1, K - 1 AA = ABS( A( I ) ) * A(k+i,k) WORK( I+K ) = WORK( I+K ) + AA S = S + AA END DO WORK( K ) = WORK( K ) + S DO J = 1, K - 1 * process S = ZERO DO I = 0, J - 2 AA = ABS( A( I+J*LDA ) ) * A(j-1,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO AA = ABS( DBLE( A( I+J*LDA ) ) ) * i=j-1 so process of A(j-1,j-1) S = S + AA WORK( J-1 ) = S * is initialised here I = I + 1 * i=j process A(j+k,j+k) AA = ABS( DBLE( A( I+J*LDA ) ) ) S = AA DO L = K + J + 1, N - 1 I = I + 1 AA = ABS( A( I+J*LDA ) ) * A(l,k+j) S = S + AA WORK( L ) = WORK( L ) + AA END DO WORK( K+J ) = WORK( K+J ) + S END DO * j=k is special :process col A(k,0:k-1) S = ZERO DO I = 0, K - 2 AA = ABS( A( I+J*LDA ) ) * A(k,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO * * i=k-1 AA = ABS( DBLE( A( I+J*LDA ) ) ) * A(k-1,k-1) S = S + AA WORK( I ) = S * done with col j=k+1 DO J = K + 1, N * * process col j-1 of A = A(j-1,0:k-1) S = ZERO DO I = 0, K - 1 AA = ABS( A( I+J*LDA ) ) * A(j-1,i) WORK( I ) = WORK( I ) + AA S = S + AA END DO WORK( J-1 ) = WORK( J-1 ) + S END DO I = IDAMAX( N, WORK, 1 ) VALUE = WORK( I-1 ) END IF END IF END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN * * Find normF(A). * K = ( N+1 ) / 2 SCALE = ZERO S = ONE IF( NOE.EQ.1 ) THEN * n is odd IF( IFM.EQ.1 ) THEN * A is normal & A is n by k IF( ILU.EQ.0 ) THEN * A is upper DO J = 0, K - 3 CALL ZLASSQ( K-J-2, A( K+J+1+J*LDA ), 1, SCALE, S ) * L at A(k,0) END DO DO J = 0, K - 1 CALL ZLASSQ( K+J-1, A( 0+J*LDA ), 1, SCALE, S ) * trap U at A(0,0) END DO S = S + S * double s for the off diagonal elements L = K - 1 * -> U(k,k) at A(k-1,0) DO I = 0, K - 2 AA = DBLE( A( L ) ) * U(k+i,k+i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * U(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO AA = DBLE( A( L ) ) * U(n-1,n-1) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF ELSE * ilu=1 & A is lower DO J = 0, K - 1 CALL ZLASSQ( N-J-1, A( J+1+J*LDA ), 1, SCALE, S ) * trap L at A(0,0) END DO DO J = 1, K - 2 CALL ZLASSQ( J, A( 0+( 1+J )*LDA ), 1, SCALE, S ) * U at A(0,1) END DO S = S + S * double s for the off diagonal elements AA = DBLE( A( 0 ) ) * L(0,0) at A(0,0) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = LDA * -> L(k,k) at A(0,1) DO I = 1, K - 1 AA = DBLE( A( L ) ) * L(k-1+i,k-1+i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * L(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO END IF ELSE * A is xpose & A is k by n IF( ILU.EQ.0 ) THEN * A' is upper DO J = 1, K - 2 CALL ZLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE, S ) * U at A(0,k) END DO DO J = 0, K - 2 CALL ZLASSQ( K, A( 0+J*LDA ), 1, SCALE, S ) * k by k-1 rect. at A(0,0) END DO DO J = 0, K - 2 CALL ZLASSQ( K-J-1, A( J+1+( J+K-1 )*LDA ), 1, + SCALE, S ) * L at A(0,k-1) END DO S = S + S * double s for the off diagonal elements L = 0 + K*LDA - LDA * -> U(k-1,k-1) at A(0,k-1) AA = DBLE( A( L ) ) * U(k-1,k-1) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA * -> U(0,0) at A(0,k) DO J = K, N - 1 AA = DBLE( A( L ) ) * -> U(j-k,j-k) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * -> U(j,j) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO ELSE * A' is lower DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) END DO DO J = K, N - 1 CALL ZLASSQ( K, A( 0+J*LDA ), 1, SCALE, S ) * k by k-1 rect. at A(0,k) END DO DO J = 0, K - 3 CALL ZLASSQ( K-J-2, A( J+2+J*LDA ), 1, SCALE, S ) * L at A(1,0) END DO S = S + S * double s for the off diagonal elements L = 0 * -> L(0,0) at A(0,0) DO I = 0, K - 2 AA = DBLE( A( L ) ) * L(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * L(k+i,k+i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO * L-> k-1 + (k-1)*lda or L(k-1,k-1) at A(k-1,k-1) AA = DBLE( A( L ) ) * L(k-1,k-1) at A(k-1,k-1) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF END IF END IF ELSE * n is even IF( IFM.EQ.1 ) THEN * A is normal IF( ILU.EQ.0 ) THEN * A is upper DO J = 0, K - 2 CALL ZLASSQ( K-J-1, A( K+J+2+J*LDA ), 1, SCALE, S ) * L at A(k+1,0) END DO DO J = 0, K - 1 CALL ZLASSQ( K+J, A( 0+J*LDA ), 1, SCALE, S ) * trap U at A(0,0) END DO S = S + S * double s for the off diagonal elements L = K * -> U(k,k) at A(k,0) DO I = 0, K - 1 AA = DBLE( A( L ) ) * U(k+i,k+i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * U(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO ELSE * ilu=1 & A is lower DO J = 0, K - 1 CALL ZLASSQ( N-J-1, A( J+2+J*LDA ), 1, SCALE, S ) * trap L at A(1,0) END DO DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) END DO S = S + S * double s for the off diagonal elements L = 0 * -> L(k,k) at A(0,0) DO I = 0, K - 1 AA = DBLE( A( L ) ) * L(k-1+i,k-1+i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * L(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO END IF ELSE * A is xpose IF( ILU.EQ.0 ) THEN * A' is upper DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE, S ) * U at A(0,k+1) END DO DO J = 0, K - 1 CALL ZLASSQ( K, A( 0+J*LDA ), 1, SCALE, S ) * k by k rect. at A(0,0) END DO DO J = 0, K - 2 CALL ZLASSQ( K-J-1, A( J+1+( J+K )*LDA ), 1, SCALE, + S ) * L at A(0,k) END DO S = S + S * double s for the off diagonal elements L = 0 + K*LDA * -> U(k,k) at A(0,k) AA = DBLE( A( L ) ) * U(k,k) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA * -> U(0,0) at A(0,k+1) DO J = K + 1, N - 1 AA = DBLE( A( L ) ) * -> U(j-k-1,j-k-1) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * -> U(j,j) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO * L=k-1+n*lda * -> U(k-1,k-1) at A(k-1,n) AA = DBLE( A( L ) ) * U(k,k) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF ELSE * A' is lower DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE, S ) * U at A(0,1) END DO DO J = K + 1, N CALL ZLASSQ( K, A( 0+J*LDA ), 1, SCALE, S ) * k by k rect. at A(0,k+1) END DO DO J = 0, K - 2 CALL ZLASSQ( K-J-1, A( J+1+J*LDA ), 1, SCALE, S ) * L at A(0,0) END DO S = S + S * double s for the off diagonal elements L = 0 * -> L(k,k) at A(0,0) AA = DBLE( A( L ) ) * L(k,k) at A(0,0) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = LDA * -> L(0,0) at A(0,1) DO I = 0, K - 2 AA = DBLE( A( L ) ) * L(i,i) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF AA = DBLE( A( L+1 ) ) * L(k+i+1,k+i+1) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF L = L + LDA + 1 END DO * L-> k - 1 + k*lda or L(k-1,k-1) at A(k-1,k) AA = DBLE( A( L ) ) * L(k-1,k-1) at A(k-1,k) IF( AA.NE.ZERO ) THEN IF( SCALE.LT.AA ) THEN S = ONE + S*( SCALE / AA )**2 SCALE = AA ELSE S = S + ( AA / SCALE )**2 END IF END IF END IF END IF END IF VALUE = SCALE*SQRT( S ) END IF * ZLANHF = VALUE RETURN * * End of ZLANHF * END