install.packages("knitr")
library(knitr)

joint_shannon_entropy <- function(alpha1, alpha2, lambda1, lambda2, delta) {
  term1 <- (alpha1 - 1) / alpha1
  term2 <- (alpha2 - 1) / alpha2
  term3 <- log(4 * alpha1 * alpha2 * lambda1 * lambda2)
  term4 <- 2 * digamma(1) 
  term5 <- digamma(alpha1 + 1)
  term6 <- digamma(alpha2 + 1)
  series_sum <- 0
  for (i in 0:10) {
    numerator <- (-1)^i * delta^(i + 1)
    denominator <- (i + 1)
    term_a <- ((1 + (-1)^(i + 1)) / (i + 2))^2
    term_b <- delta * ((1 + (-1)^(i + 2)) / (i + 3))^2
    series_sum <- series_sum + (numerator / denominator) * (term_a + term_b)
  }
  term7 <- (1/4) * series_sum
  term1 + term2 - term3 - term4 + term5 + term6 - term7
}

alpha1_values <- c(0.1, 1, 2.5, 5)
alpha2_values <- c(0.1, 1, 2.5, 5)

delta_values  <- c(-0.9, 0.1, 0.9)

lambda_pairs <- list(
  c(0.7, 0.9),   
  c(1.2, 1.4)    
)

for (lp in lambda_pairs) {
  lambda1 <- lp[1]; lambda2 <- lp[2]
  for (delta in delta_values) {
    M <- outer(
      alpha1_values, alpha2_values,
      Vectorize(function(a1, a2) joint_shannon_entropy(a1, a2, lambda1, lambda2, delta))
    )
    dimnames(M) <- list(paste0("alpha1=", alpha1_values),
                        paste0("alpha2=", alpha2_values))
    cat(sprintf("\n--- Grid: delta = %g, lambda1 = %g, lambda2 = %g ---\n",
                delta, lambda1, lambda2))
    print(round(M, 3))
  }
}
<pre>An Error occurred while handling another error:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/ErrorHandler.php(136): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/ErrorHandler.php(135): yii\web\ErrorHandler-&gt;renderException()
#3 [internal function]: yii\base\ErrorHandler-&gt;handleException()
#4 {main}
Previous exception:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(656): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/models/Files.php(696): yii\base\Application-&gt;end()
#3 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/controllers/FilesController.php(484): faravaghi\filemanager\models\Files-&gt;getFile()
#4 [internal function]: faravaghi\filemanager\controllers\FilesController-&gt;actionGetFile()
#5 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/InlineAction.php(57): call_user_func_array()
#6 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Controller.php(180): yii\base\InlineAction-&gt;runWithParams()
#7 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Module.php(528): yii\base\Controller-&gt;runAction()
#8 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Application.php(103): yii\base\Module-&gt;runAction()
#9 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(386): yii\web\Application-&gt;handleRequest()
#10 /var/www/html/prof-homepages/backend/web/index.php(16): yii\base\Application-&gt;run()
#11 {main}</pre>