"The cluster, 
as found in the word 
and in similar position in the other words. While reading the first letter 
in such a cluster it is always pronounced like 
(ṭ) and the following 
as it is, that is trill. Here is pronounced as (ṭr) as in the English word, ‘attraction’. This is an important one to be noted by the learners of Tamil with respect to reading/speaking and writing.”
I cannot believe a Tamil manual used at a university like UT, Austin is teaching this fundamentally wrong pronunciation. The first ற் should be pronounced as alveolar t. It is true that nowadays some youngsters erroneously bring in retroflexion in pronouncing this cluster. But that cannot be taught as the standard pronunciation.
Here are some Youtube clips with the occurrence of ற்ற் cluster, where the first ற் is clearly not a retroflex.
https://www.youtube.com/watch?v=yfHbOXpk-GQ (released 1965)
https://www.youtube.com/watch?v=3YwtyCxkgQc (released ca. 2013?)
Here is a clip with the pronunciation of ற்ற் clusters with retroflexion by Sikkil Gurucharan (ca. 2012?).
I hope the author/s go through the manual carefully and correct it.
Regards,
Palaniappan