LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ 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 principal minor of order i is not positive, 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)
Definition cblat2.f:3285
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
subroutine stpsv(uplo, trans, diag, n, ap, x, incx)
STPSV
Definition stpsv.f:144
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