Make usage page consistent with example1.dat

Author: usamamuneeb

docs/usage/index.md

@@ -24,18 +24,18 @@ As an example, we will solve the following specific CLP in this section as a gen
 $$
 \begin{aligned}
 &\boldsymbol{A}=\left(\begin{array}{cccc}
-10 & 4 & 4 & 0 \\
-0 & 0 & 0 & -8 \\
-0 & -8 & -8 & -2
+-10 & -4 & -4 & 0 \\
+0 & 0 & 0 & 8 \\
+0 & 8 & 8 & 2
 \end{array}\right), \quad \boldsymbol{b}=\left(\begin{array}{c}
-48 \\
--8 \\
-20
+-48 \\
+8 \\
+-20
 \end{array}\right), \quad \boldsymbol{c}=\left(\begin{array}{c}
--11 \\
+11 \\
 0 \\
 0 \\
-23
+-23
 \end{array}\right) \\
 &\text { J.f }=3, \quad \mathrm{~K} . \mathrm{s}=2,
 \end{aligned}
@@ -144,22 +144,22 @@ import sdpap
 import numpy as np
 
 A = np.array([
-    [10, 4, 4, 0],
-    [0, 0, 0, -8],
-    [0, -8, -8, -2]    
+    [-10, -4, -4, 0],
+    [0, 0, 0, 8],
+    [0, 8, 8, 2]
 ])
 
 b = np.array([
-    [48],
-    [-8],
-    [20]
+    [-48],
+    [8],
+    [-20]
 ])
 
 c = np.array([
-    [-11],
+    [11],
     [0],
     [0],
-    [23]
+    [-23]
 ])
 
 K = sdpap.SymCone(s=(2,))
@@ -172,46 +172,46 @@ The output will be as below:
 
 ```
 ---------- SDPAP Start ----------
-SDPA start at [Sun Jun  5 10:43:05 2022]
+SDPA start at [Sun Jan 26 20:41:26 2025]
 NumThreads  is set as 12
 Schur computation : DENSE 
-Entering DMUMPS driver with JOB, N, NZ =  -2           0              0
 Converted to SDPA internal data / Starting SDPA main loop
    mu      thetaP  thetaD  objP      objD      alphaP  alphaD  beta 
- 0 1.0e+04 1.0e+00 1.0e+00 -0.00e+00 -1.20e+03 1.0e+00 9.1e-01 2.00e-01
- 1 1.6e+03 0.0e+00 9.4e-02 +9.23e+02 -7.51e+01 2.3e+00 9.6e-01 2.00e-01
- 2 1.7e+02 6.4e-17 3.6e-03 +2.80e+02 +3.74e+01 1.3e+00 1.0e+00 2.00e-01
- 3 1.8e+01 6.4e-17 1.5e-17 +7.70e+01 +4.19e+01 9.9e-01 9.9e-01 1.00e-01
- 4 1.9e+00 7.2e-17 7.5e-18 +4.57e+01 +4.19e+01 1.0e+00 9.0e+01 1.00e-01
- 5 1.9e-01 7.3e-17 1.1e-15 +4.23e+01 +4.19e+01 1.0e+00 1.0e+00 1.00e-01
- 6 1.9e-02 6.4e-17 2.2e-17 +4.19e+01 +4.19e+01 1.0e+00 9.0e+01 1.00e-01
- 7 1.9e-03 6.3e-17 2.1e-15 +4.19e+01 +4.19e+01 1.0e+00 1.0e+00 1.00e-01
- 8 1.9e-04 8.3e-17 3.0e-17 +4.19e+01 +4.19e+01 1.0e+00 1.0e+00 1.00e-01
- 9 1.9e-05 7.9e-17 1.5e-17 +4.19e+01 +4.19e+01 1.0e+00 9.0e+01 1.00e-01
-10 1.9e-06 6.5e-17 1.6e-15 +4.19e+01 +4.19e+01 1.0e+00 9.0e+01 1.00e-01
+ 0 1.0e+04 1.0e+00 1.0e+00 -0.00e+00 +1.20e+03 1.0e+00 9.1e-01 2.00e-01
+ 1 1.6e+03 0.0e+00 9.4e-02 +8.39e+02 +7.51e+01 2.3e+00 9.6e-01 2.00e-01
+ 2 1.7e+02 2.3e-16 3.6e-03 +1.96e+02 -3.74e+01 1.3e+00 1.0e+00 2.00e-01
+ 3 1.8e+01 2.9e-16 2.2e-17 -6.84e+00 -4.19e+01 9.9e-01 9.0e+01 1.00e-01
+ 4 1.9e+00 2.6e-16 1.8e-15 -3.81e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
+ 5 1.9e-01 2.7e-16 7.5e-18 -4.15e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
+ 6 1.9e-02 2.8e-16 7.5e-18 -4.19e+01 -4.19e+01 1.0e+00 9.0e+01 1.00e-01
+ 7 1.9e-03 2.9e-16 1.2e-15 -4.19e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
+ 8 1.9e-04 2.8e-16 2.2e-17 -4.19e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
+ 9 1.9e-05 2.7e-16 7.5e-18 -4.19e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
+10 1.9e-06 2.9e-16 3.7e-18 -4.19e+01 -4.19e+01 1.0e+00 1.0e+00 1.00e-01
 
 phase.value  = pdOPT     
    Iteration = 10
-          mu = +1.9180668442023463e-06
-relative gap = +9.1554458700816248e-08
-        gap  = +3.8361319951718542e-06
-     digits  = +7.0383205007122669e+00
-objValPrimal = +4.1900003836133735e+01
-objValDual   = +4.1900000000001739e+01
-p.feas.error = +7.2685417628031834e-15
-d.feas.error = +1.5063505998114124e-12
-total time   = 0.016597
-  main loop time = 0.016594
-      total time = 0.016597
+          mu = +1.9180668442023158e-06
+relative gap = +9.1554505917001858e-08
+        gap  = +3.8361336223147191e-06
+     digits  = +7.0383202767393023e+00
+objValPrimal = -4.1899996163866383e+01
+objValDual   = -4.1900000000000006e+01
+p.feas.error = +3.5690251616723664e-14
+d.feas.error = +3.5527136788005009e-15
+total time   = 0.003271
+  main loop time = 0.003268
+      total time = 0.003271
 file  check time = 0.000000
 file change time = 0.000003
 file   read time = 0.000000
 Converting optimal solution to CLP format
-SDPA end at [Sun Jun  5 10:43:05 2022]
+SDPA end at [Sun Jan 26 20:41:26 2025]
 Start: getCLPresult
 Making result infomation...
-/opt/anaconda3/envs/py310/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1265: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
-  warnings.warn("k >= N - 1 for N * N square matrix. "
+(Re)calculating feasibility errors for CLP converted solution.
+/opt/anaconda3/lib/python3.12/site-packages/sdpap/sdpaputils.py:114: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
+  eig = eigs(mat.toarray(), k=1, which='SM',
 ========================================
  SDPAP: Result
 ========================================
@@ -221,15 +221,15 @@ Making result infomation...
      frvMethod = split
   domainMethod = none
    rangeMethod = none
-     primalObj = -4.1900000000001739e+01
-       dualObj = -4.1900003836133735e+01
-    dualityGap = +9.1554458700816248e-08
-   primalError = +1.5063505998114124e-12
+     primalObj = +4.1900000000000006e+01
+       dualObj = +4.1899996163866383e+01
+    dualityGap = +9.1554505917001858e-08
+   primalError = +3.5527136788005009e-15
      dualError = +0.0000000000000000e+00
-   convertTime = 0.000262
-     solveTime = 0.018632
-retrievingTime = 0.000003
-     totalTime = 0.020157
+   convertTime = 0.000166
+     solveTime = 0.011680
+retrievingTime = 0.000002
+     totalTime = 0.011902
 ---------- SDPAP End ----------
 ```