 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ spptrf()

 subroutine spptrf ( character UPLO, integer N, real, dimension( * ) AP, integer INFO )

SPPTRF

Purpose:
``` SPPTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A stored in packed format.

The factorization has the form
A = U**T * U,  if UPLO = 'U', or
A = L  * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] AP ``` AP is REAL 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. See below for further details. On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.```
Further Details:
```  The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14
a22 a23 a24
a33 a34     (aij = aji)
a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]```

Definition at line 118 of file spptrf.f.

119 *
120 * -- LAPACK computational routine --
121 * -- LAPACK is a software package provided by Univ. of Tennessee, --
122 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
123 *
124 * .. Scalar Arguments ..
125  CHARACTER UPLO
126  INTEGER INFO, N
127 * ..
128 * .. Array Arguments ..
129  REAL AP( * )
130 * ..
131 *
132 * =====================================================================
133 *
134 * .. Parameters ..
135  REAL ONE, ZERO
136  parameter( one = 1.0e+0, zero = 0.0e+0 )
137 * ..
138 * .. Local Scalars ..
139  LOGICAL UPPER
140  INTEGER J, JC, JJ
141  REAL AJJ
142 * ..
143 * .. External Functions ..
144  LOGICAL LSAME
145  REAL SDOT
146  EXTERNAL lsame, sdot
147 * ..
148 * .. External Subroutines ..
149  EXTERNAL sscal, sspr, stpsv, xerbla
150 * ..
151 * .. Intrinsic Functions ..
152  INTRINSIC sqrt
153 * ..
154 * .. Executable Statements ..
155 *
156 * Test the input parameters.
157 *
158  info = 0
159  upper = lsame( uplo, 'U' )
160  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
161  info = -1
162  ELSE IF( n.LT.0 ) THEN
163  info = -2
164  END IF
165  IF( info.NE.0 ) THEN
166  CALL xerbla( 'SPPTRF', -info )
167  RETURN
168  END IF
169 *
170 * Quick return if possible
171 *
172  IF( n.EQ.0 )
173  \$ RETURN
174 *
175  IF( upper ) THEN
176 *
177 * Compute the Cholesky factorization A = U**T*U.
178 *
179  jj = 0
180  DO 10 j = 1, n
181  jc = jj + 1
182  jj = jj + j
183 *
184 * Compute elements 1:J-1 of column J.
185 *
186  IF( j.GT.1 )
187  \$ CALL stpsv( 'Upper', 'Transpose', 'Non-unit', j-1, ap,
188  \$ ap( jc ), 1 )
189 *
190 * Compute U(J,J) and test for non-positive-definiteness.
191 *
192  ajj = ap( jj ) - sdot( j-1, ap( jc ), 1, ap( jc ), 1 )
193  IF( ajj.LE.zero ) THEN
194  ap( jj ) = ajj
195  GO TO 30
196  END IF
197  ap( jj ) = sqrt( ajj )
198  10 CONTINUE
199  ELSE
200 *
201 * Compute the Cholesky factorization A = L*L**T.
202 *
203  jj = 1
204  DO 20 j = 1, n
205 *
206 * Compute L(J,J) and test for non-positive-definiteness.
207 *
208  ajj = ap( jj )
209  IF( ajj.LE.zero ) THEN
210  ap( jj ) = ajj
211  GO TO 30
212  END IF
213  ajj = sqrt( ajj )
214  ap( jj ) = ajj
215 *
216 * Compute elements J+1:N of column J and update the trailing
217 * submatrix.
218 *
219  IF( j.LT.n ) THEN
220  CALL sscal( n-j, one / ajj, ap( jj+1 ), 1 )
221  CALL sspr( 'Lower', n-j, -one, ap( jj+1 ), 1,
222  \$ ap( jj+n-j+1 ) )
223  jj = jj + n - j + 1
224  END IF
225  20 CONTINUE
226  END IF
227  GO TO 40
228 *
229  30 CONTINUE
230  info = j
231 *
232  40 CONTINUE
233  RETURN
234 *
235 * End of SPPTRF
236 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
subroutine sspr(UPLO, N, ALPHA, X, INCX, AP)
SSPR
Definition: sspr.f:127
subroutine stpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPSV
Definition: stpsv.f:144
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