SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AP( * )
* ..
*
* Purpose
* =======
*
* DSPTRF computes the factorization of a real symmetric matrix A stored
* in packed format using the Bunch-Kaufman diagonal pivoting method:
*
* A = U*D*U**T or A = L*D*L**T
*
* where U (or L) is a product of permutation and unit upper (lower)
* triangular matrices, and D is symmetric and block diagonal with
* 1-by-1 and 2-by-2 diagonal blocks.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
* On entry, the upper or lower triangle of the symmetric matrix
* A, packed columnwise in a linear array. The j-th column of A
* is stored in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
* On exit, the block diagonal matrix D and the multipliers used
* to obtain the factor U or L, stored as a packed triangular
* matrix overwriting A (see below for further details).
*
* IPIV (output) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D.
* If IPIV(k) > 0, then rows and columns k and IPIV(k) were
* interchanged and D(k,k) is a 1-by-1 diagonal block.
* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, D(i,i) is exactly zero. The factorization
* has been completed, but the block diagonal matrix D is
* exactly singular, and division by zero will occur if it
* is used to solve a system of equations.
*
* Further Details
* ===============
*
* 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
* Company
*
* If UPLO = 'U', then A = U*D*U', where
* U = P(n)*U(n)* ... *P(k)U(k)* ...,
* i.e., U is a product of terms P(k)*U(k), where k decreases from n to
* 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
* defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
* that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
* ( I v 0 ) k-s
* U(k) = ( 0 I 0 ) s
* ( 0 0 I ) n-k
* k-s s n-k
*
* If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
* If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
* and A(k,k), and v overwrites A(1:k-2,k-1:k).
*
* If UPLO = 'L', then A = L*D*L', where
* L = P(1)*L(1)* ... *P(k)*L(k)* ...,
* i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
* n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
* defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
* that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
* ( I 0 0 ) k-1
* L(k) = ( 0 I 0 ) s
* ( 0 v I ) n-k-s+1
* k-1 s n-k-s+1
*
* If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
* If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
* and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
$ KSTEP, KX, NPP
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
$ ROWMAX, T, WK, WKM1, WKP1
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
EXTERNAL LSAME, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSPR, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRF', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
IF( UPPER ) THEN
*
* Factorize A as U*D*U' using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2
*
K = N
KC = ( N-1 )*N / 2 + 1
10 CONTINUE
KNC = KC
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 110
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( AP( KC+K-1 ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value
*
IF( K.GT.1 ) THEN
IMAX = IDAMAX( K-1, AP( KC ), 1 )
COLMAX = ABS( AP( KC+IMAX-1 ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
ROWMAX = ZERO
JMAX = IMAX
KX = IMAX*( IMAX+1 ) / 2 + IMAX
DO 20 J = IMAX + 1, K
IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
ROWMAX = ABS( AP( KX ) )
JMAX = J
END IF
KX = KX + J
20 CONTINUE
KPC = ( IMAX-1 )*IMAX / 2 + 1
IF( IMAX.GT.1 ) THEN
JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 )
ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K-1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K - KSTEP + 1
IF( KSTEP.EQ.2 )
$ KNC = KNC - K + 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the leading
* submatrix A(1:k,1:k)
*
CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
KX = KPC + KP - 1
DO 30 J = KP + 1, KK - 1
KX = KX + J - 1
T = AP( KNC+J-1 )
AP( KNC+J-1 ) = AP( KX )
AP( KX ) = T
30 CONTINUE
T = AP( KNC+KK-1 )
AP( KNC+KK-1 ) = AP( KPC+KP-1 )
AP( KPC+KP-1 ) = T
IF( KSTEP.EQ.2 ) THEN
T = AP( KC+K-2 )
AP( KC+K-2 ) = AP( KC+KP-1 )
AP( KC+KP-1 ) = T
END IF
END IF
*
* Update the leading submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
* Perform a rank-1 update of A(1:k-1,1:k-1) as
*
* A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
*
R1 = ONE / AP( KC+K-1 )
CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
*
* Store U(k) in column k
*
CALL DSCAL( K-1, R1, AP( KC ), 1 )
ELSE
*
* 2-by-2 pivot block D(k): columns k and k-1 now hold
*
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Perform a rank-2 update of A(1:k-2,1:k-2) as
*
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
*
IF( K.GT.2 ) THEN
*
D12 = AP( K-1+( K-1 )*K / 2 )
D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
D11 = AP( K+( K-1 )*K / 2 ) / D12
T = ONE / ( D11*D22-ONE )
D12 = T / D12
*
DO 50 J = K - 2, 1, -1
WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
$ AP( J+( K-1 )*K / 2 ) )
WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
$ AP( J+( K-2 )*( K-1 ) / 2 ) )
DO 40 I = J, 1, -1
AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
$ AP( I+( K-1 )*K / 2 )*WK -
$ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
40 CONTINUE
AP( J+( K-1 )*K / 2 ) = WK
AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
50 CONTINUE
*
END IF
*
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K-1 ) = -KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
KC = KNC - K
GO TO 10
*
ELSE
*
* Factorize A as L*D*L' using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2
*
K = 1
KC = 1
NPP = N*( N+1 ) / 2
60 CONTINUE
KNC = KC
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 110
KSTEP = 1
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( AP( KC ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value
*
IF( K.LT.N ) THEN
IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 )
COLMAX = ABS( AP( KC+IMAX-K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
ELSE
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value
*
ROWMAX = ZERO
KX = KC + IMAX - K
DO 70 J = K, IMAX - 1
IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
ROWMAX = ABS( AP( KX ) )
JMAX = J
END IF
KX = KX + N - J
70 CONTINUE
KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
IF( IMAX.LT.N ) THEN
JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 )
ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) )
END IF
*
IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
*
* interchange rows and columns K and IMAX, use 1-by-1
* pivot block
*
KP = IMAX
ELSE
*
* interchange rows and columns K+1 and IMAX, use 2-by-2
* pivot block
*
KP = IMAX
KSTEP = 2
END IF
END IF
*
KK = K + KSTEP - 1
IF( KSTEP.EQ.2 )
$ KNC = KNC + N - K + 1
IF( KP.NE.KK ) THEN
*
* Interchange rows and columns KK and KP in the trailing
* submatrix A(k:n,k:n)
*
IF( KP.LT.N )
$ CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
$ 1 )
KX = KNC + KP - KK
DO 80 J = KK + 1, KP - 1
KX = KX + N - J + 1
T = AP( KNC+J-KK )
AP( KNC+J-KK ) = AP( KX )
AP( KX ) = T
80 CONTINUE
T = AP( KNC )
AP( KNC ) = AP( KPC )
AP( KPC ) = T
IF( KSTEP.EQ.2 ) THEN
T = AP( KC+1 )
AP( KC+1 ) = AP( KC+KP-K )
AP( KC+KP-K ) = T
END IF
END IF
*
* Update the trailing submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = L(k)*D(k)
*
* where L(k) is the k-th column of L
*
IF( K.LT.N ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
*
* A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
*
R1 = ONE / AP( KC )
CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
$ AP( KC+N-K+1 ) )
*
* Store L(k) in column K
*
CALL DSCAL( N-K, R1, AP( KC+1 ), 1 )
END IF
ELSE
*
* 2-by-2 pivot block D(k): columns K and K+1 now hold
*
* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
* where L(k) and L(k+1) are the k-th and (k+1)-th columns
* of L
*
IF( K.LT.N-1 ) THEN
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
*
D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
T = ONE / ( D11*D22-ONE )
D21 = T / D21
*
DO 100 J = K + 2, N
WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
$ AP( J+K*( 2*N-K-1 ) / 2 ) )
WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
$ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
*
DO 90 I = J, N
AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
$ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
$ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
90 CONTINUE
*
AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
*
100 CONTINUE
END IF
END IF
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -KP
IPIV( K+1 ) = -KP
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
KC = KNC + N - K + 2
GO TO 60
*
END IF
*
110 CONTINUE
RETURN
*
* End of DSPTRF
*
END